Math Revisited: Complex Analysis (1)

In advanced physics, including plasma physics and quantum mechanics, the importance of complex numbers and analysis can't be over emphasized. Consider the diagram shown with three complex numbers identified.   To extend the generality of complex numbers and enhance their applicability, it's useful to write them in what's called "polar form". 

A critical part is finding the angle shown, referred to as the argument. We can see from the diagram that the angle Θ may be found using:

arctan (y/x) = arctan(3/4) = 36.8 deg

Thus, Θ = 36.8 degrees is the argument


Now, any complex number (x + iy) may be written in polar form:

  x + iy = r(cos (Θ) + isin(Θ))


To get r:

r = [x² + y²]^1/2 =  [4² + 3²]^1/2 = [25]^1/2 = 5

Therefore we may write:  

 (x + iy) = 5(cos (36.8) + isin(36.8))

Note there is also the abbreviated function (based on the combo of sine and cosine):

cis (Θ) = cos (Θ) + isin(Θ)

So we can finally write:

C = r cis(Θ) = 5 cis (36.8)

We look now at the vectors A and B, which we’ll henceforth call z1 and z2 to be consistent with complex notation. Our eventual goal will be to find the resultant, which will come in the next installment. In the meantime we will be working toward showing the multiplication and division of two complex forms, call them z1 and z2: 

e.g.  [z1 + z2]

From the diagram:

A= z1 = -2 + 2i

B = z2 = -2 -3i

So: z1 = x1 + iy1

And  arg(z1) = arctan(y1/x1) = arctan (-2/2) = arctan(-1)

So (Θ1) = -45 degrees = -π /4

Now find r1:

r₁ =[x₁² + y₁²]^1/2 =  [1 + 1]^1/2 = Ö2   Therefore:


z1 = Ö2 (cos(-45) + isin(-45)) = Ö2 cis(-45)

We now turn to the vector B which is:  z2 = x2 + iy2= -2 -3i

then: arg(z2) = arctan(y2/x2) = arctan (-3/-2) = arctan (3/2) = 56.3 deg

While:

r₂ = [x₂² + y₂²]^1/2 =  [(-2)² + (-3)²]^1/2 = [13]^1/2 =  3.6


Therefore:

z2 = 3.6(cos(56.3) + isin(56.3) = 3.6 cis(56.3)

Now, how do we obtain the complex product: [z1•z2]?

We have that:

[z1•z2] = (z1•z2) cis(arg(z1) – arg(z2))

But:

(z1•z2) = Ö2 (3.6) = 5.1

And:

arg(z1) – arg(z2) = (-45) – (56.3) = -101.3

so that:


[z1•z2] = 5.1 cis(-101.3) = 5.1 (cos (-101.3) + isin(-101.3))


[z1•z2] = 5.1((-0.195) + i(-0,98))


[z1•z2] = 0.99 + 0.98i

To get the resultant: z1 + z2 = z3:

A + B = z1 + z2 =[ (-2 + 2i) + (-2 – 3i)] = -4 –i

In any case: x3 + iy3 = - 4 – i

Complex Division:

Let's say we want to divide:

z1 = Ö2(cos(-45) + isin(-45)) = Ö2 cis(-45)

by

z2 = 3.6(cos(56.3) + isin(56.3)) = 3.6 cis(56.3)

In all such cases of complex division we require that the z, r in the denominator not be zero.

Thus:

(z₁/z₂) =  (r₁ cis(q₁)/ r₂ cis(q₂)) = (r₁/ r₂) cis (q₁ –  q₂)

 

Now: (r1/ r2) = (1.414/ 3.6) = 0.39

And we saw previously:

(q₁ – q₂)   = arg(z₁) – arg(z₂) = (-45) – (56.3) = -101.3

Thus, the basic procedure for division entails dividing the lengths (r’s) and subtracting the angles    (q₁ – q₂).

So:
(z₁/ z₂) =  0.39 (cos (-101.3) + isin(-101.3))

= 0.39((-0.195) + i(-0.98)) = -0.07 + 0.38i
 

What about?   (1 + i)  ¸  Ö3  – i

The first order of business is to get dividend and divisor each into polar form, specifically as a (cis) function. 

Then (1 + i) = z₁  = x1 + iy1, so arg(z₁ ) = arctan (y1/x1)
Further:

arctan (y1/x1) = arctan (1/1) = arctan (1) so q₁ = 45 deg

What about r1?

r1= [1² + 1²]^1/2 =  Ö2 = 1.4

so z₁ = 1.4 [cos (45) + isin(45)] = 1.4 cis(45)
 

Now: z₂ = Ö3  – i

So arg(z₂) = arctan(y₂/x₂) =  arctan(-1/ Ö3) so q₂

= (-30 deg)

r₁/r₂ = Ö2/ 2 

Then subtract angles: [(q₁ – q₂) ] = {(45 deg) – (-30 deg)} = 75 degrees

So the end result of the division is:

(z₁/z₂)  =  Ö2/ 2   cis(75) = Ö2/ 2  {cos(75) + isin(75)}

= 0.707{cos(75) + isin(75)}

Since cos(75) = 0.258 and sin(75) =0.966, we have:

(z₁/z₂)   = 0.707[(0.258) + i(0.966)] = 0.183 + 0.683i

Another very convenient way to express complex numbers is in the exponential form.

Thus, we can write: cos(q) + isin(q) = r exp (iq )

Thus, the previous numbers we divided (z₁ and z₂) may be expressed:

z₁ = Ö2 [cos (45) + isin(45)] = Ö2 exp (i p/4)

z2 = 2[cos(-30) +isin(-30)] = 2 exp(i (-p/6))

Practice Problems:

1) Express each of the following end results in the form:   r exp(iq):

a) (2 + 3i)(1 – 2i)

b) (1 + i) (1- i)

c) (1 + Ö-3)²

2) Plot the results of (b) and (c) on the same Argand diagram and obtain the resultant. Check

Trump's First 100 Days: The Disaster Wrought By An Unqualified Fool - As We Expected

"Hey, my biggest achievement after one hundred days is playing more golf and taking more time off than any other president! Whaddya want from me? Real work?"

The verdict is in and it doesn't take another 100 days, or 300 days to ascertain once and for all that Trump is every bit the incompetent buffoon, and entitled egomaniac most of us expected. His results after the one hundred day mark? Essentially zilch. Promises kept? None, as in zero. Worse, on the 100th day (today) this asshole has us on the brink of  a possible nuclear war, or at least a war that will claim an untold number of lives.

Not so startling, this twit now agrees with most of the American people that he was woefully unprepared for the job, vastly more than any other incoming president.

So let's cut the crap and the media's newfound worship of this dunderhead. What exactly is he best known for after 100 days? We can list his "milestones" below:

- More time spent away from the White House than any other occupant, approaching 30 percent of his total 'in office time' and nearly all at his Florida hacienda called Mar-a -Lago.

- More golf played  on office time than any other president, much of it also at Mar-a-Lago

- Most security breaches while taking time off, including the fiasco at Mar- a-Lago while dining with the Japanese PM, Shinzo Abe. There receiving a classified report of a North Korean missile launch at his dining table - while being photographed by guests with cell phone cameras. More recently firing off 59 cruise missiles with the Chinese President  Xi Jinping present. Ignoring him to make an attack, and basically pissing away any diplomatic capital.

Trump subsequently revealed that it took a conversation with the president of China to realize that the situation on the Korean peninsula was “not so easy.”  DOH! You didn't know North Korea is a nuclear state and you can't just lob 59 cruise missiles into Pyongyang.

- Most inept executive orders (36) ever written, most at the level of a ten -year old trying to spoof a president. Nearly all now moribund or challenged by the courts - the latest to do with blocking the funding for sanctuary cities.

- Total abject failures in two of the biggest promises to his supporters: the building of the border wall which Mexico was to pay for,  but the costs now to be transferred to American taxpayers. Oh, and the failed repeal of Obamacare.  Both moves had to be withdrawn recently to avoid a government shutdown.

- An incompetent tax cut bill which will blow the deficit to new dimensions, exceeding $7 trillion - none of which has any remote chance of being paid for.  (It has about as much chance being paid for by "economic growth" as Roswell aliens showing up on Infowars to snatch Alex Jones before his next broadcast.)

- More unvetted appointments, including of Steve Bannon, and Michael Flynn, than any other president.

- More conflicts of interest and business entanglements - violating the emoluments clause of the Constitution - than any other president in history.

-  More anti-agency appointments, naming heads of agencies like the EPA, antithetical to their existence, than any other president.

- The first president in history to communicate almost exclusively via a cartoon medium (twitter) who remains in campaign mode even as he does.

- The most outrageous presidential liar ever, including Richard Milhous Nixon.

Recall that during the campaign Trump boasted incessantly of being the "best president ever" and "never leaving the White House".   But  getting a streak of no BS honesty this week with a reporter from Reuters , Trump had a different assessment of the presidency. He whined:

“I love my previous life. I had so many things going. This is more work than in my previous life, I thought it would be easier. I thought it was more of a …business-type scene.  I’m a details-oriented person. I think you’d say that, but I do miss my old life. I like to work .....but this is actually more work.”

In other words, he openly admitted he's not cut out for the job, has no business even changing the toilet paper in the guest rooms, far less writing any executive orders or issuing threats against a nuclear -armed state. He's basically an overgrown, entitled baby in diapers who has to be mollycoddled and "managed" by his team of patient handlers. Off the record, they all agree 'the Donald' is akin to an overgrown infant and to get things done you have to catch him in his good moments and implement unconscious 'time outs' (unconscious for him) to get him to even begin to do what you want. (And keep all sharp objects out of his reach, especially when in the midst of a temper tantrum - when some outside force blocks his actions.)

As I'd written before this wasn’t the first time that Trump copped to the job being harder than he anticipated. In November, NBC News reported that Trump had told former House speaker Newt Gingrich that “This is really a bigger job than I thought.” (Gingrich’s response? “…good. He should think that.”) Then there are individual issues. “Nobody knew health care could be so complicated,” he said at one point.

But this is balderdash, because any candidate with sense and a modicum of knowledge knew that, including Jill Stein and Hillary Clinton. Even Gary Johnson the Libertarian candidate had more of an insight than Trump into the complexity of U.S. healthcare.  Trump tried to substitute his bluster and bombast for actual knowledge which never works. So what Trump really meant to say is that "I DIDN'T KNOW health care could be so complicated 'cause I am basically a know nothing!"

So this 100 day "did nothing" mark elicits again the question of why the blustery braggart doesn't settle in to work at the White House. Well,  because the cockeyed, weasel bastard hates to work at governance! He'd rather loaf in the WH, or lollygag in campaign mode in front of his groupies.   That is what he considers "working"- being in campaign (or twitter) mode 24/ 7.

Recall the Donald has never really worked at a real job  such as one most of his followers have: little control over conditions and outside interference.  All he's known is being top banana in his family branding business (who could order anyone to do his bidding) and being a NY real estate weasel who used bankruptcy (of multiple casinos) to make more $$$.  Hell, five days into his presidency he  even bawled that the job had  turned out to be too much for him to deal. Too many details, too much stuff to master for an extensive government bureaucracy.  Hence, his penchant for becoming a cable TV and tweet addict. See e.g.

http://theconcourse.deadspin.com/donald-trump-stunned-to-learn-presidency-is-an-actual-j-1792215349

Excerpt:

“Being president is harder than Donald Trump thought,” begins the article, neatly capturing the blithe, criminal ignorance that characterizes both Trump himself and the many dozens of millions of morons who thought he should be the leader of the free world. Yes, being the president is a harder job than Donald Trump would expect, because Donald Trump had never previously held an actual job, because actually, spending your inheritance on a succession of failed cons is not an actual job."

In this light, Trump taking hundreds of hours off to hob nob at Mar-a-Lago makes sense. If the actual job is really too much for his three year old emotional brain then of course he'd want to escape the responsibilities as much as possible. So hence he'd hightail it to Mar-a-Lago where he could be in a comfortable environment more resonant with his weasel business man shtick.

Make no mistake, Trump's  recent Reuters comments reinforce his earlier ones and disclose an even greater element of surprise (and more than a hint of bafflement)  that having responsibility for the welfare of 320 million people entwined . It is as if the "dog" finally caught the bus he was chasing for months and doesn't know what to do with it. "Jeez, that thing is a lot lot bigger than I thought! Now what?"

This is the bristling brat to whom we've entrusted our futures, or rather the 25 % of Trump voters who elected this turd and now have the sane population hostage under his derelict purview..

Trump is right about one thing, in his case the one hundred day mark is "meaningless."  Because in 1,000 more days it's doubtful he'll have accomplished anything beyond the puddle of doggie lickspittle he already has. The message is clear: He should have remained in his weasel business where the extent of destruction left in his wake would have been vastly reduced from what he's inflicting on this nation.

For Donald Trump The 25 Million Souls Of Seoul Are Expendable In A North Korea Conflict

"There's gonna be a major, major conflict! And I'm sorry about Seoul! Tough luck there! But their sacrifice'll make me look more presidential!"

As The Financial Times reported yesterday, the bellicose Trump bunch wants Jimmy Carter to stay out of any peace -making deals with North Korea. This despite the war rhetoric reaching new heights. In a Reuters interview yesterday, the egomaniacal baboon- in- chief grunted, then growled: "We're heading for a major, major conflict with North Korea. Absolutely!'

And as I posted earlier, this hotheaded maggot doesn't possess the skill, background historical knowledge or intelligence to steer clear of a conflagration with a nuclear state that would unleash a pitiless assault not on the U.S. (which is too far away to strike) but on its South Korean ally. Specifically, the 25 million citizens of Seoul, the "most densely populated city on Earth" according to one general reporting to the Senate Armed Services Committee. 

Does Trump care one whit about any of those innocent civilians when Kim Jung-Un unleashes a barrage of tens of thousands of shells from 4,000 artillery pieces? Of course not! For him, they are in part or all at once expendable pawns on the grand "chessboard" on which he's playing the mighty "sheriff" of the world. Having been stoked to the brim with his Syria cruise missile strike, he now believes he can unleash a more devastating one on North Korea. But he's a damned fool and idiot because it is the millions of Seoul who will pay.

Yes, the U.S. would ultimately come out ahead but it would be a Pyrrhic victory. Meaning the costs would exceed the brief moment of victory, mainly for the South Koreans who'd have to body bag the dead (what body parts they could find) and rebuild their devastated city. And I'm assuming here no nuclear weapons would be used which might be pie in the freaking sky.

The Bulletin of the Atomic Scientists (the same group that operates the ‘doomsday clock’ and has now moved it two and a half minutes to midnight) estimates the North possesses twenty weapons in the 10 – 20 kiloton range or roughly the same magnitude of yield as the atomic bomb that reduced Hiroshima to ashes in 1945.

 Would they actually use it? 12 days ago in an interview with CBS’ Ben Stacy, the North Korean Defense minister when asked about any response if the U.S. made a pre-emptive strike didn’t bandy words: he said there would be a total nuclear retaliatory strike.  How serious should he be taken?

 First, they are estimated to be at the most primitive, early stages of A-bomb development. According to the most recent Physics Today (April issue) their existing weapons would deliver a yield of roughly 0.08 kt/ kilogram. That is yield per actual bomb mass.  This means they’d need to be able to launch or carry (via plane) at least 100 kg of bomb mass to get an 8 kiloton A -blast, or 200 kg to get a 16 kt yield.   This is feasible with their existing bombers and might also be with their most advanced (solid fuel) missile. However, none of these weapons could reach the U.S. presently – though they could wreak havoc on Seoul, South Korea – with the potential to obliterate over 20 million people as well as all U.S. troops stationed in S. Korea.

Clearly, this is a cost and risk no sane leader would visit on a vulnerable ally. And yet Trump, who fancies himself Dictator of the world, has no qualms about doing so - from his rhetoric- nor any sympathy for the South Koreans who will be caught in the line of fire as a possible mini nuclear holocaust plays out.

 Second, that they would make such a retaliatory strike is beyond question. As numerous commentators have pointed out – e.g. Sue Mi Terry - former CIA North Korean analyst, they’d regard it as a matter of survival. Much like Castro during the Cuban Missile Crisis in October, 1962 who told former Kennedy Defense Secretary Robert McNamara in 1992 (during a 30 yr. reunion) he’d have launched 93 nuclear-armed IRBMs had the U.S. invaded. (JFK was being pushed at the time to bomb and invade Cuba by his Joint Chiefs of Staff).

 When asked in an MSNBC interview 2 weeks ago if Trump would actually attack North Korea, Sue Terry responded:

 “I can’t see him following through on this and that is the problem with the brinksmanship policy. Because you’re putting yourself in a bind. You will either have to back down and lose credibility or you are stuck on a ledge with a military option which is very, very risky.

 North Korea is not Syria. It’s not Afghanistan. It’s going to have very devastating consequences.  North Korea will retaliate to any kind of military option. They will retaliate against South Korea given seventy percent of its ground forces are deployed within 100 kilometers (60 miles) of the DMZ.  And there’s twenty thousand U.S. military in South Korea and twenty million people in Seoul”

 But if his words yesterday are taken seriously, and there's no reason to believe this insane maggot is bluffing - given his ego-- then the South can expect ferocious blowback and devastation. Trump then is not  a rational actor like JFK was during the Cuban Missile Crisis. He’s an imperious,  impetuous braggart and clueless numbskull narcissist not even aware of recent history. He operates according to an unstable  “high conflict” personality which one psychologist (Bill Eddy) has said makes him a “more dangerous politician  than Adolf Hitler”.  In other words, unlike with JFK, we can only assume Trump will make exactly the opposite decision in a nuclear game of high stakes poker.

The sad and sorry Truth is that Donald Trump cares no more about the lives of those millions living in Seoul, than he did about the gassed kids in Syria prompting his cruise missile attack. Those kids were  merely transient blips on his cable TV screen arousing a transient emotion then a brief brain "rupture"  which saw that if he made a bold response he might profit. And profit he did! The media backed off its close scrutiny and criticism and suddenly saw this POS as "presidential" .

Expect him to play "double or nothing" with the Korean standoff, but we know already who the real losers of this charlatan's  game will be.
----------------------------------------------
To understand the full background, read this handy collection of some of the world's leading experts published by the Bulletin of the Atomic Scientists. Feel free to forward links to others:

North Korea in the news again: a time to reflect, edited by Dan Drollette

Will South Korea go nuclear? Robert Einhorn, Duyeon Kim

North Korea's nuclear weapons: What now? Roundtable with Shen Dingli, Chung-in Moon, Andrei Lankov

Hecker assesses North Korean hydrogen bomb claims, Siegfried Hecker

The North Korea that can say no, Bruce Cumings

Getting North Korea wrong, Bruce Cumings

Math Revisited: Linear Algebra (Dimension)

The concept of 'dimension' in linear algebra basically has direct applicability for all finite dimensional vector spaces, up to some limit, n. There also occur infinite dimensional vector spaces for which it's possible to provide an infinite basis, but these are more peculiar to specialized applications as in advanced quantum theory (e.g. Hilbert spaces as infinite dimensional vector spaces). We will eschew this special version until a much later time, next year, when we survey some topics in advanced quantum mechanics.

Meanwhile, if V is some vector space having a basis consisting of 'n' elements, we say that n is the dimension of V.

If V consists of 0 alone, then V does not have a basis, and one says V has dimension 0.

As another generic example, let the vector space Rⁿ     have dimension n over R, and the vector space C ⁿ   have dimension n over C. More generally, for any field F, the vector space F ⁿ  has dimension n over F. One can therefore say that the n vectors:

(1, 0.........), (0,1...........0), (0, 0, 1..........0),.........(0, ............0, 1)

form a basis of  Fⁿ  over F.

Then the dimension of the vector space V over F is denoted by: dim_F V or more simply, dim V.

Associated with these considerations is the concept of a maximal set of linearly independent elements of a vector space. Then let: v1, v2, v3.......vn be linearly independent elements of a vector space V, then the elements w, v1, v2, .....vn are linearly independent and v1, v2, v3.......vn for a set of maximally independent elements.

Ancillary or Auxiliary Theorems:

1) Given V is a vector space and one basis has m elements and another basis has n, then m = n.

2) Let V be a vector space and {v1, v2, v3.......vn } be a maximal set of linearly independent elements of V, then {v1, v2, v3.......vn } is a basis of V.

3) Let V be a vector space consisting of n elements. Let W be a subspace which does not consist of zero alone. Then W has a basis and the dimension of W is less than or equal to n.

4) Let V be a vector space over the field F and let U,W be subspaces. If: U + W = V and if U/\W = {0} then V is the direct sum of U and W. (Note: /\ denotes intersection)

5) If V is a finite dimensional vector space over F, and is the direct sum of subspaces U, W then:

dim V = dim U + dim W


Problems:

1) Let V = R² and let W be the subspace (2,1). Let U be the subspace generated by (0, 1). Show that V is the direct sum of W and U.

2) Prove theorems (1) - (4)

3) what is the dimension of the space of 2 x 2 matrices? Give a basis for this space.

Parsing The "High Conflict Personality" - Does Trump Qualify?

"Hey! I'm a high conflict guy! Deal with it!"


Evidently, there's a new subclass of psycho that has been flying under the radar and which needs to be integrated into the maelstrom of mentally unstable misfits, including malignant narcissists and authoritarians. That is, the "high conflict personality".   According to Bill Eddy, one of the 'high priests' of this specialty area:

"Narcissistic high conflict people like being unpredictable and beholden to no one. They hate rules and feel superior to them. So being consistent feels confining and they generally like to keep everyone else guessing. It’s rule by personality, not by logic or law. Sociopathic high conflict people will say (lie) and do whatever is convenient or appealing at the time, without conscience or consistency. They like instability because it keeps everyone else off-balance. This especially can be demonstrated with a lack of loyalty to his associates, discarding them at will and turning his attention to new shiny objects and people (wives, for example)."


This brand of looneytune sounds like Donald Trump and with good reason: it basically nails his unstable, predatory shtick to a tee. The question is: Can such an unbalanced person govern with any degree of efficacy - and the answer nearly 100 days into this misbegotten presidency appears to be a resounding 'no'.  Nothing Trump has done thus far, from his half-assed, juvenile "executive orders", to his impetuous cruise missile strike on Syria, to his bellicose rhetoric in confronting Kim Jong Un over the North Korea issue, shows he possesses the leadership mettle to inspire any confidence. He is more a buffoon, and totally ignorant at that. Hence, his only ploy or play is to try to keep others off balance. Or to use the phrase of one HuffPost contributor: "A mad monkey with a gun loose on the street".

He does this by constant appeal to conflict.  Is this a dangerous and demented shtick? Yes it is. According to Bill Eddy who wrote the definitive book on high conflict personalities,  Trump is  “…potentially the most dangerous person in politics since Adolf  Hitler.”  Got that? The most dangerous politico since Adolf Hitler.

 In an interview with a San Diego journalist,  referencing the quote, Eddy was asked: "Clearly a terrifying opinion — yet we do remember the ardent hordes of Hitler’s followers. Can you elaborate?"

Eddy replied:

"There are similarities regarding how he connects with his followers, so that they become more passionately connected when he and they are criticized. It strengthens their bond. It’s a major part of my book. In a nutshell, it’s because he uses emotions instead of logic, and uses repetition on a greater scale than most politicians since Hitler. He has conditioned his followers to his simple arguments from day one. Hitler was known for spewing ten times as many words as any other politician, and he used modern media for voice (daily speeches  on the radio) and face (movies of him speaking at his huge rallies). These emotional media projecting face and voice are much more powerful on our brains than printed words."

In other words, Trump has gone for the "lizard brains" of his doting followers and enslaved them wholesale. This is something that neuroscientist Robert Ornstein once raised, in his book, 'The Evolution of Consciousness' and later works, on how easily the limbic system is activated and overtakes reason. In fact, we are talking about an entire brain complex, referred to as the "R-complex" - for example by Carl Sagan in his book, 'The Dragons of Eden'.

The main aspect to note concerning the R-complex and its dynamics is that primitive emotion rules, and logic is far removed from any integration. It's as if reason doesn't exist. This is also why it's impossible to get through rationally to the Trump groupies and voters. Their brains are pitched to the emotional centers of the R-complex not to the neocortex higher thought centers. You could as well be
arguing with a Komodo Dragon.

It's unsettling enough to find these properties in ordinary citizens, but  absolutely terrifying for them  to be associated with leaders whose fingers are on the nuclear football.  This is because the lizard brain, i.e. R-complex, doesn't weigh responses in proportion to stimuli. If buzzed by a fly it will reach for a sledge hammer, and if a hornet's nest has been built on a deck it will take out the flame thrower. If  the house burns down "them's the breaks".  This is why having an unhinged "high conflict" nut like Trump at the helm for this Korean crisis ought to keep everyone up at night - even the fool who scribbled a recent NY Times piece to the effect there's nothing to worry about.  "Despite the showdown there won't be a war."  Will you place a bet on that, fool?

When the same SD journalist asked Eddy:  "Do you really believe that, per his leadership, 'sooner or later we will have a war on our hands?'”

Eddy replied by quoting a statement from his book:

“He tends to inspire violence and lack of restraint — which leads to lack of physical restraint, which leads to organized aggressive behavior — which leads inevitably to war. He will “split” the world into allies and enemies. We will have more chaos than we do now in the Middle East — and on American soil. Friends and family members will start hating each other, and school children will become disrespectful and violent towards people who look different from them.”

When  Eddy was asked if we ought to scared, that is those of us with functional higher brain centers, he replied:

"Yes. Moderate Republicans  may limit his damage and keep him tied up in procedural  knots but it won't be enough."

What we need is the Dems to take over the House next year and begin impeachment proceedings as soon as feasible. It is clear we can't depend on the Reepos to do their jobs. They have too much invested in this maniac to help push through their radical agenda. Stay tuned.

Math Revisited: Linear Algebra (3)

We now look at orthonormal bases. The term sounds esoteric but many would have encountered it before either in general physics or in advanced (AP) physics courses taken in high school. This would be in conjunction with the dot product of vectors, such as illustrated in Fig. 1.

Basically, in some Euclidean (straight line, 3D) system of coordinates, two vectors are considered orthogonal if their inner product is zero, as shown. The geometric properties here are assumed to be on the basis being orthonormal, i.e. composed of pairwise perpendicular vectors with unit length. Thus, in the figure shown, vectors A and B meet this condition (and the computation is shown for the vectors A, B) as do the vectors B and C. Then the vectors A, B, C meet the condition for an orthonormal basis. There are proofs available but those are beyond the scope of this blog.

Now, in applications of this concept, what the student is usually asked to do (say in his linear algebra course) is find an orthonormal basis for a "subspace" of R ³  (e.g. applied to a Cartesian space of three dimensions) which is generated by specified sets of vectors.

Example:

Find an orthonormal basis for the subspace of  R ³ generated by the vectors (1, 1, -1) and (1, 0 ,1).

We let A = (1, 1, -1) and B = (1, 0, 1)

The orthonormal basis for A is just:

A/ ‖A ‖ = (1, 1, -1)/ {1²  + 1 ²   + (-1) ² } =    (1, 1, -1)/  Ö 3

The orthonormal basis for B is:

B/ ‖B ‖ = (1, 0, 1)/ {1  + 0   + (1)²  } =

(1, 0 , 1)/ Ö2


Of course, the beauty of linear algebra is that it can be generalized to Euclidean spaces beyond the mundane 3, hence we can look at subspaces in  R ⁴, generated by sets of vectors (v1, v2, v3, v4).

Example:

Find an orthonormal basis for the subspace of R ⁴ , generated by the vectors:

A = (1, 2, 1, 0) and B = (1, 2, 3, 1)

For A we have the orthonormal basis:

A/ ‖A ‖ = (1, 2, 1, 0)/ {1 ²  +  2 ²  + 1 ²  +  0 ² } =

(1, 2, 1, 0)/ Ö6

For B we have:

B/ ‖A‖ = (1, 2, 3, 1)/ {1 ² + 2 ² +  3 ² + 1 ²} =

(1, 2, 3, 1)/Ö 15


Practice Problems:

(1) Find the orthonormal basis for the subspaces of  R ³  generated by the vectors:

A = (2, 1, 1) and B = (1, 3, -1)

(2) Find the orthonormal basis for the subspaces of  R ⁴ generated by the vectors:

A = (1, 1, 0, 0)

B = (1, -1, 1, 1)

C = (-1, 0, 2, 1)

(3) Find an orthonormal basis for the subspace of the complex space  C ³ generated by the vectors:

A = (1, -1, i)

and

B = (i, 1, 2)

Thoughts On The Closure Of ALL EXPERTS

All Experts was the free question and answer service that for 19 years provided users with answers to their questions in a vast variety of subject areas from biology, medicine, geology, religion, politics, and psychology to astrophysics and physics.  Since late 2004 I participated in the Astronomy forum - this before there was All Experts (it was known then as 'Experts Central') then three years later joined the astrophysics expert forum, as well as the atheist -agnostic forum.

As per a notification sent out a week ago, the site shut down from April 24th. Anyone trying to access it via Google or the url will simply be shown a terse message noting the 19-year run and that "more than two million" questions were answered.

Why the closure, we can't say. Experts were simply sent the brief notice with no further explanation. One reason is that perhaps there wasn't the money available to support the financial maintenance and even a putatively "free site" needs money to operate - if only to continue to support nearly 500 different forums, responses and further exchanges. Not to mention tracking vacation dates and experts' statistics in their profiles.

But let's be clear the site wasn't always smooth sailing and for experts often a source of frustration. After eight years answering astronomy questions, for example, I had to go to the option of  "no ratings allowed"  - ratings being given in the several areas of: politeness, knowledge, clarity of response, and timely response.    What constantly irritated me was that so few questioners knew how to give an appropriate rating, especially to knowledge and clarity of response.  For example, they'd deliver a '10' in knowledge but then a  '6' for clarity of response. But how would one be cognizant that the knowledge aspect was as high as 10 when they believed the clarity was only at a 6? It was preposterous.  

I tried to attach the proviso that if the answer still appeared to be abstruse, or too difficult, the questioner could reframe the question or make it more specific. (That was another gripe - often the questions were so poorly worded one couldn't make any sense of them and had to ask the person to please rewrite it, try to be more specific or to clarify a term used).   Often there was no further question, so the expert is left with just the crummy rating - and bear in mind we aren't getting paid a damned cent so the ratings comprise our only "earnings".

I retained the ratings for the astrophysics largely because I'd become a lot more rigid in what categories of question I accepted. If it seemed that bare balderdash or ignorance was the theme I delivered a polite rejection response.  Or, if there was the slightest divergence from what astrophysics included. For example, questions to do with meteorology, say the formation of "Moon dogs" or "halos"  forming around the Moon.   While perhaps making it as an astronomy question - ice crystals forming in the atmosphere - it doesn't make the cut for astrophysics. 

As I made clear in subsequent rejection letters the topics for astrophysics included: star formation and evolution, the HR diagram, stellar models, stellar spectra, star cluster dynamics, galactic cluster properties, Hubble diagram, expansion of the universe, solar flares, sunspot properties and coronal mass ejections as well as prominences, plasma waves and instabilities. 

Apart from the poorly worded or misdirected questions there were the many from crackpots. These were most often poorly or semi-educated people who were perhaps above average in intelligence, had a bit of a "brain storm" and fancied themselves a new Einstein for having found some seemingly novel relationship. One of these was a guy in his fifties or sixties, admittedly having "little education" who came up with the notion of "Earth flares", i.e. analogs to solar flares but occurring on Earth.

After several fruitless exchanges where I emphasized there was no observational basis for his entity, namely any magnetic signature for rapid release, he finally agreed to go to another expert. He just couldn't admit his own "thesis" was cockeyed and - in essence - pseudo science.

Another pair of crackpots (from the Middle East) insisted they had overturned Kepler's 2nd law of planetary motion, the "equal area" law.  One of them actually wrote:

"Kepler's area law says r*Vp=Ct.  Newton's universal attraction force says this force is radial F=Fr and a perpendicular force (Fp)  to the radial, also a side force component does not exist.  So: m*dVp/dt=Fp=0.  Then dVp/dt=0 and with integration we get Vp=Ct.     f Vp=Ct is correct, elliptical orbits theory has to be modified to a new theory, new math. And the motion equation should be r=-4*t^2+4*t*T-4*T^2/6 .This equation does not indicate an ellipse but a parabolic vortex spiral."

For his benefit I made several points:

1) He never expressed the 2nd law in proper form and indeed it makes no sense at all as portrayed.

2) He never defined 'C', and whether it is the same as h .  He doesn't say so one can't just assume it.

3) His equation r*dVp = Ct = 0 is incorrect. The Areal law in most concise form should read:

dA/dt = h z^ / 2

In addition,  he failed to distinguish between a real force and a non-existent force, i.e. perpendicular to the orbit (since the gravitational force of interaction acts through the mass centers) so mg = 0 i.e. weightlessness applies.

In effect, his integration result, e.g. Vp=Ct  is spurious .

After much back and forth in the rejection regime he rendered a final assertion he believed would force me to go to his side.  This was after I pointed out that the very fact spacecraft had landed successfully on Mars, the Moon, Venus etc. showed the Kepler law in its given form had to be correct. However, unwilling to accept this he wrote:

"Anyhow,  sending celestial probes to the Moon,or Mars or to any body will still be successful even if the orbits are triangular. It does not depend on  the form of the planet's trajectory . It is controlled from the earth."

In other words, he's saying you just need to fire the rocket into space, and the human controllers on Earth (like at Johnson Space Center) will do the rest,  to "steer" it to the destination.    I had to summon all my will power to avoid calling him a total nutcase, but again we must allow common courtesy even for kooks.

Then there was the Canadian "Bree"  (and several others with different identities or names) who sought to browbeat me into accepting a novel presentation of units which "proved" one or more of her crackpot ideas. She wrote, as "mathematical proof":


1)     Speed of Light  =  orbit Velocity^2  /  Pi       (Enormously Important Equation)

2)     Speed of Light  =  Acceleration of Gravity  x  orbit Time

3)     Speed of Light  =  Acceleration of Gravity  x  (earth Volume  /  orbit Circumference x Tau2)

4)     Speed of Light  =  Acceleration of planet earth in orbit  x  Diameter of orbit

5)     Speed of Light  =  Density of earth  x  orbit Velocity x 21  

None of which made any sense in terms of actual physical units and dimensions as I tried to show her, to no avail.  Her baloney was a perfect example of similar piffle cited by Charles Seife in his superb book:  'Proofiness: How You're Being Fooled by the Numbers.'   Seife decried the tactic of using numbers not just to lie but to baffle the susceptible and gullible with bullshit.  He referred to a common failing of most people unversed in math to be hoodwinked merely because some form of math or numbers are interjected into arguments.  Not just using numbers to bolster one's argument. In his words, to use fake numbers to prove falsehoods and to seek to prove something is true - even when it's not- is one of the most egregious forms of  intellectual  fraud. But "Bree" believed herself to be an incomparable genius of the first order.

The first thing any first year Physics student learns - or should - is to check a formula or claimed relationship for consistency of the physical units used. If their combination doesn't yield any recognized physical quantity - defined as part of the standards and constants published yearly by the American Institute of Physics - then the results are balderdash, pure and simple. Let's take her example (4) where:

Speed of Light  =  Acceleration of planet earth in orbit  x  Diameter of orbit

What are the units here? The speed of light c is in meters per second, or m/s. The acceleration a is given in terms of  m/ s ²  and the diameter of any orbit (any orbit) is a length -  so given in meters (m). Then multiplying them together one gets:

(m/s)  =  ?  (m/ s ²) (m)  =  m² / s ²

Which is emphatically not the same as the units for velocity of light! Let's also check her "enormously important equation" (1): Speed of Light  =  orbit Velocity^2  /  Pi      

Again, we examine the product of the units to see if there's a physically meaningful quantity that results or if this girl is just seeing things, or seeing what she wants to see.  We have the speed of light on one side (units m/s ) and on the other the orbit velocity squared:

(m/s)²    =  m² / s ²


Note that the presence of pi is neither here nor there as it's dimensionless.       "Bree" leaves herself open on so many fronts it's essentially laughable, and in ordinary discourse she'd be put down as a dummy or dupe, but as experts  we had to strive for politeness. This irrespective of what mutation of knowledge we were confronted with. (Including one Saudi character nicknamed "Amtry" who tried to argue atheists were less than moral because they objected to have prostitutes flogged, or those women seeking abortions. 

The thing that really bothered me in all this kerfuffle with the Canuck lass(-es) was, not so much the snarkiness and bogus certitude of these young women, but their failure to recognize that the combination of units yields zilch. A squared velocity is simply NOT the same as a velocity. Both girls - if indeed there are two - would merit an F in any physics class of mine.  Worse, if I could I'd deliver an F-minus because of the lack of critical thinking.

Sadly, if this girl or girls are infected by this stupidity many others might be too. Who knows how many physics students have gone on to the website referenced and then belabored their profs with this bullshit, wasting all kinds of time?  But one thing I've learned is you can't argue with ignorant students or ignorant people who are already committed to believe what they want. In this case, because the "maths' seems to work out (at least in their heads) they believe it like a sacred book.

One would have thought they'd back off but that's not the nature of crackpots.  This was even after I had written in one rejection:.  Btw, anyone can concoct and hurl together dissimilar units to make them appear valid, but garbage in = garbage out. The sure sign of pseudo-scientific claptrap and bunkum is when one reads:

"The earth in orbit is electrical, the Tesla is present due to the earth moving through the aether.  When you multiply earth Acceleration by the Diameter as well as the hidden Tesla you get the Velocity of light and its CORRECT UNITS."

Any physics student of mine that would scrawl such drivel would get an F - for the semester! An orbit being electrical? Moving through the "ether"? The ether doesn't even exist - the Michelon -Morley experiment showed the concept of ether is redundant.  The rest is pure poppycock. The waste of a mind is a terrible ting and this nonsense is an enormous waste of a mind - as well as the minds led to buy into it."

 At least 5 months passed before the following bit of bafflegab gibberish (under the header 'Sound math')  arrived in my inbox, doubtless compliments of the selfsame Canadian twit, now using the name "Brianna":

Our SOLARMath equations work with "Kilograms" ( http://members.shaw.ca/warmbeach/INDEX3.htm )

But because Avogadro's number is atoms per mole which leads to atoms per "gram", prior to commencing the equation below, we need to convert the initial Mass of the earth to grams.

The Equation is;

The Mass of the earth divided by Gravity^4  x  Acceleration of earth in orbit  =  Avogadro's number.

5537831004648121688015772977.2114697 grams  /  (9.8)^4  =  600392689687827221617859.5082045622325038

600392689687827221617859.5082045622325038  x  .001003207246557  =  602318297074676474879.204

I replied with a customized rejection and was as polite as possible, emphasizing that the astrophysics forum (at least for my domain of questions) had to meet at least one of a set  of content criteria for consideration as a legit astrophysics query. She then fired back "Are you drunk? Of course it's not astrophysics, it's math!"    But any serious inquirer in astrophysics - or math- would know the multi-digit explication as shown would instantly identify  a hardcore crank -  not to mention the units being all wrong.

Then there was Steve Torchinsky  (an "engineer turned radio astronomer") who "torched" me  ("so-called astronomy expert") for advising an Indian engineering student  who wanted to transfer into astrophysics that it might be "too hard" a path to successfully negotiate. Especially going from engineering to highly abstract (relatively!) astrophysics courses like stellar modeling, evolution and plasma theory.  Torchinsky blabbered on about the hundreds who "made it" (say in radio astronomy) but never referenced my presumption that one had to get through a very difficult undergrad and graduate series of courses first. A battery that many engineers - including my mechanical engineering brother in law- have said would have been their undoing. Could he have made it? Maybe. Would he have? I don't know and neither does Torchinsky.

Apart from these issues there've always been the niggling critiques (in ratings sites)  of All Experts, i.e. they "don't know enough about my specific problem" or "they take too much time off etc. WTF do you expect for a FREE service? All of which shows me that the widespread trope that "people devalue and dismiss anything given for free" is probably true. So, as they say - let the naysayers and complainers carry  on now that All Experts is gone. I am sure Google will be able to answer all their specific questions - if they just take enough time and get through the first gazillion irrelevant pages!