Examples Of Problems To Use In Active Learning For Mathematics Teaching

"Active learning" has now come to the fore in much post-secondary mathematics teaching. This has been referenced in a recent article in the Notices of the American Mathematical Society ('What Does Active Learning Mean For Mathematicians?, Vol. 64, No. 2, p. 124). The definition given is:

"Active learning refers to classroom practices that engage students in activities such as reading, writing, discussion or problem solving, that promote higher order thinking."

In fact, a major component is inquiry -based learning. This entails "students spending class time working on problem sets individually or in groups, presenting solutions and or proofs to the class, and receiving feedback from peers and faculty"

An important point to note, is that unlike inquiry-based learning in physics (as I discussed in my March 27 post), the courses integrating AL are not based on pure, unguided student discovery. Rather, faculty design a series of carefully structured activities for individuals, pairs, or small groups - as well as some for whole classes.  Below I give some ideas for individual and paired AL problem solving I myself have used over the years in assorted college math classes I've taught:

1) Abstract Algebra:

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

Example:

(G, o) = (R1 +)

(H, o) = (R*, ·)

Take f = the exponential function, so f(x) = exp (x), f(y) = exp(y)

Then: f(x + y) = exp(x + y) = exp(x) exp(y) = f(x) f(y)

Or:

H = R* = {x Î R: x  not equal 0}


And: exp R -> R* so exp(x + y) = exp(x) exp(y)


Def.: Isomorphism: An isomorphism of G onto H [(G, o), (H, o)] is a bijective homomorphism.


Example: H = P = {x Î R: x > 0}         (P, x)


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

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

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

2. Discrete Mathematics:

i) Show that (p ® q)  «  (~q ® ~p)  is a tautology.                                       

    (ii) Let x  Î { 2, 3, 4} and y Î {12, 16}.  Let the propositional function

P(x, y) be the statement “x is a factor of y”.  Write the following  propositions using conjunctions and disjunctions and determine the truth value of each.

            (i) "x $y P(x, y)              (ii) $y "x P(x, y)                                          

(iii)   Show that log n! = O (n log n).                                                                

(iv)     Let f(x) = 2x³ + 3x –1 and g(x) =  log x. 

Find the least integer n such that:
 (fg)(x)=          O(xⁿ)                                                                                          

(v)Use mathematical induction to prove that

       (1x2) + (2x3) + (3x4) + … + n(n + 1)  =  n(n +1)(n +2)/3.                     

       (vi) Prove that if n is an integer  and n³ + 5 is odd, then n is even using:

                        (a) indirect proof         (b) proof by contradiction.   

3) Linear Algebra:

I) Show that the matrix M =

(1 + i.....2)
(2..........5i)

is not Hermitian

ii) Determine whether the matrix Y =

(1.....(1+ 1i).......5)
((1- i).......2... ...i)
(5..........-i...........7)


is Hermitian or not.

iii) Determine whether the matrix, X =

(-i...1)
(1.....i)

is unitary or not.

iv) Let A and B be 2 x 2 Hermitian matrices. Show that (A + B) is Hermitian.

4) Analytic Geometry:

Two ellipses are graphed on the same axes as shown below:

Obtain the analytic equation for each ellipse, and in particular, show how they are related to one another.

5) Calculus:

i) Find the residue for f(z) =  3 exp (z)/  z ⁴

ii) Find all the residues at those singular points inside the circle  ÷ z  ÷     =    2  

For:  f(z) =   z ²    / ( z ⁴  - 1)

iii) ∫ _-¥  ^¥  (1 + x² )  dx / 1 + x⁴     =   ?

iv)  ∫ _-¥  ^¥    x  dx / (x²   - 2x + 2)  =  ?

6) Differential equations:

i)  100 gallon tank is full of pure water. Let pure water run into the tank at the rate of 2 gals/ min. and a brine solution containing 1/2 lb. of salt run in at the rate of 2 gals/min. The mixture flows out of the tank through an outlet tube at the rate of 4 gals/min. Assuming perfect mixing, what is the amount of salt in the tank after t minutes?

ii) A block of mass m = 2.0 kg rests on a smooth horizontal surface attached to a spring. The spring has the property that it is stretched 0.05 m by a force of 10 N. If the block is displaced 0.05 m from the equilibrium position and released, find: the frequency and period of the motion.

iii) Consider the diagram of a rocket's trajectory shown below:

Where the launch angle is denoted q_o, and the final x, y- coordinates (e.g. at impact) are shown as the terminal point of the path.

Use appropriate differential equations to find the  x- and y-coordinates of the terminal point on the trajectory for a rocket launched at an angle of 80 degrees with an initial velocity of 100,000 f/s if the air resistance is 0.01mv. Also find the value of x and y after 10 seconds.

Note: The layout for such active math learning was generally in a lab setting using tables at which students could work collaboratively, with me and at least one assistant in attendance to provide insights if needed.  Generally, the students (based in Barbados) were able to work through each numbered set or problem in one hour or less. After, there was time for discussion - given a 90 minute total class time - with followup inquiry problems to be worked out.

The same layout can also be used to conduct Math Olympiads, with three or four student teams competing against each other. Many other permutations on active learning are discussed in the AMS Notices article, and ought to provide lecturers much room for more fully engaging their students.

Trump In Psycho Redux Tries To Link Susan Rice To His "Wiretaps"

Recall that exactly a month ago, the orange-haired baboon who sits in the Oval office  launched a series of tweets that began at 5.35am, scribbling in one : “Terrible! Just found out that Obama had my ‘wires tapped’ in Trump Tower just before the victory. Nothing found. This is McCarthyism!”

He followed up in the next half-hour with a string of deranged tweets claiming Obama had "defied a court rejection to tap his office", and invited a “good lawyer” to make a case against the alleged process.  Very swiftly, lawyer Ari Melber on MSNBC skewered Trump's psychotic delusions noting that if there was a wiretap it was more plausibly ORDERED by a judge on account of suspicion of criminal activity or collusion - possibly with a foreign power.  So in fact, Trump's tweets - as opposed to any victimization - could actually have been open admissions that he has something to be guilty about, if such wiretaps were actually FISA -warrant approved.

Melber's views were given major heft at the first and only televised meeting of the House Intelligence Committee  when FBI Director James Comey openly acknowledged before that the FBI was conducting an active investigation into the ties between the Trump campaign and Russians blew all Trumps' leak paranoid fantasies into a cocked hat.

Since then Trump and his devoted lackeys and trained monkeys - like Devin Nunes - have tried relentlessly to provide cover for the baboon -in -chief's insanity. Nunes, most notably and recently, acted in collusion with two 'dummy' sources (Michael Ellis and Ezra Cohen-Watnik)  to use ostensible "secret" documents that would shed light on Nunes' House investigation. In fact the documents were spurious, merely laundered, raw intel that Bannon and Cohen -Watnik massaged to make it look to dummy Devin as if he had something.

And like a trained monkey, Nunes obliged and played the role Bannon assigned him to a tee. In front of the press corps Nunes claimed he  "recently confirmed" that  "on numerous occasions the intelligence community had  incidentally collected info about U.S. citizens involved in the Trump transition."  

Of course, as that Nunes' trick has been exposed and shot down, Trump and his house of clowns had to find a new avenue to keep the spotlight off the Russian connection and collusion issue. So who to pick next? Why, the dignified former Obama national security adviser forced to wear a veritable crown of thorns during the Benghazi spectacle  - as Reepo lackeys rode their high horses to ever higher hypocrisy. (It was Republican budget cuts that sliced into embassy security around the world, including in Barbados).

Thus Trump asserted in a semi-coherent delusional state earlier today that he thought that the former national security adviser Susan E. Rice may have committed a crime by seeking the identities of Trump associates who were swept up in the surveillance of foreign officials by American spy agencies and that other Obama administration officials may also have been involved. The little stooge went on to blabber:

"I think it’s going to be the biggest story. It’s such an important story for our country and the world. It is one of the big stories of our time.”

Actually, you psycho dunce,, its a non-story. Merely another attempt at deflection to drive attention away from the core Russian hacking and collusion thread. Contrary to Trump's fulsome rubbish, current and former intelligence officials from both Republican and Democratic administrations have said that nothing they have seen led them to believe that Ms. Rice’s actions were unusual or unlawful. When Americans are swept up in surveillance of foreign officials by intelligence agencies, their identities are supposed to be obscured, but they can be revealed for national security reasons, and intelligence officials say it is a regular occurrence.

Former CIA Assistant Director John McLaughlin, interviewed last night, said:

"Susan Rice was merely doing her job. Occasionally these reports refer to foreigners contacting Americans about something so the national security adviser has the option of asking the FBI, or the National Security Agency if it can be justified to let her know who those people are- and often to protect them. Sometimes you want to go to an American and say, 'you should know who you've been I contact with'.   This is done quietly and not spread around the government, And people that make decisions for unmasking someone have a very high bar for doing it, it isn't done casually,  So I think she was basically just doing her job as she should have been doing it."

So again we see Trump's paranoia - without a whit of evidence - getting the better of him as he accused Ms. Rice of a "crime". Thus, no surprise that a spokeswoman for Ms. Rice, Erin Pelton, said in an email, “I’m not going to dignify the president’s ludicrous charge with a comment.”  As well she shouldn't, because we have run amok Reepos - like Rand Paul and Lindsey Graham- who have no sense or shame, and will make hay out of nothing. After all, look how they went after Ms. Rice in the Benghazi spectacle - when real security norms were transgressed - by the REEPOS! (Trey Gowdy inadvertently blowing the cover of a CIA source. With goons like these who needs enemies?)

In an interview on Tuesday with MSNBC, Ms. Rice said: “The allegation is that somehow the Obama administration officials utilized intelligence for political purposes. That’s absolutely false.”

The only one coming off as untrustworthy here is the baboon- in -chief, an asshole who has a history of promising to produce evidence to back up his unverified claims, and failing to do so.  Indeed, he didn't even make it clear what crime he was accusing Ms. Rice of committing. (As I noted earlier its legal and not unusual for a national security adviser to request the identities of Americans who are mentioned in intelligence reports.)

What we have then is a three ring circus, with the Orange Baboon the ringmaster, Devin 'Clouseau'  Nunes first clown, and the GOP 'wanna get Rice' assholes the junior clowns.  All of this done in an effort to launch a wild goose chase tangent to try to misdirect the intelligence committee investigation into the Russian ties to the Trumpies.. And in the process present laundered "evidence" that hey, Trump was really surveilled by that angry black man and former Prez Obama.

But in this light John McLaughlin's take is perhaps most spot on regarding Trump's original tweet:

"When he did that tweet I immediately called a friend and said 'we're going to have weeks and possibly months of turmoil for what was a foolish action.'  This has deepened the partisanship and made it much harder for this administration to do anything else it wants to do, such as health care and tax reform."

Bingo!  We can be sure if the past is any guide, this clown show and circus will continue, and the next proof will be if Susan Rice is actually called to testify in either one or both intel committees.


Update:   Some good news since yesterday's post. Bannon has been forced out of the National Security Council, most likely by Lt. Gen. McMaster putting his foot down. Don't believe the bull pocky that it was "just a realignment".. Yeah, right! And Area 51 aliens were behind it. Also, Devin Nunes stepped down from his Chairman position of the House Intel Committee, but not without blowing more smoke out of his ass. He claimed "leftwing activist groups with the Office of Congressional Ethics" forced him out. Which is the epitome of rank delusion and playing the victim card.  No, no one was to blame but the little Trump and Bannon pawn himself. But it is also interesting that the punk stepped down after Bannon was removed from the NSC. Connection? You bet!  Bannon overplayed his cards and both he and his pawn Nunes paid the price.

Climate Change Alarms Ring Louder With New Findings

Divers investigate coral die off on Great Barrier Reef.


Even as climate denier and EPA head Scott Pruitt  set to obliterate that agency's ability to keep us informed, new findings reveal that the sheer dimension of the climate change crisis upon us is rapidly defying  hyperbole.   I have tried in previous posts to alert blog readers, followers to the magnitude of the crisis (also referenced Naomi Klein's must -read book, 'This Changes Everything') but words begin to fail as the latest revelations have been made from three academic sources.

This is in stark contrast to the codswallop of Scott Pruitt, Trump and their motley crew of climate change deniers who've left a  bewildered global public in the wake of their deception, lies and distortions. Because as I've shown repeatedly,, the science behind climate change is rock solid with the impacts – observed on every ecosystem on the planet – are occurring faster in many parts of the world than even the most pessimistic scientists predicted.   In this regard, I probably became even more pessimistic after seeing the state of dying corals off Barbados south coast several years ago while in a submersible craft, e.g.

The degree of bleaching and coral death was appalling, given how many other aquatic species depend on the corals' health. Janice even expressed the unspoken fear of many accompanying us in the voyage of 'Atlantis II': "I wonder if any sea life will be here when we come back again."

 Given this, it’s logical to assume life on Earth – I'm talking about the millions of species that cohabit our little ball of rock in space with us – would be impacted. But for me, Janice and many others who looked at what's going on, it's unnerving to discover that this is no longer about just polar bears; coral reefs and sea turtles. No,  it's about practically everything – including us- the human inhabitants and would be stewards of this world.

Anyway, three recent studies have illustrated just how widespread climate change’s effect on life on our planet has already become.  A team led by Bret Scheffers of the University of Florida just published a  study in Nature Climate Change in February that found that 47 percent of land mammals and 23 percent of birds have already suffered negative impacts form climate change. In all, nearly 700 species in just these two groups are flagging under climate change, according to this research.

Scheffers is also the lead author of a landmark Science study from last year that found that current warming (just one degree Celsius) has already left a discernible mark on 77 of 94 different ecological processes, including species’ genetics, seasonal responses, overall distribution, and even morphology – i.e. physical traits including body size and shape.

A few examples to fix ideas: woodland salamanders are shrinking in the Appalachian Mountains; the long-billed, Arctic-breeding red knot is producing smaller young with less impressive bills leading to survival difficulties. Meanwhile, the alpine chipmunks of Yellowstone National Park have actually seen the shape of their skulls change due to climate pressure.. And as reported in the Denver Post two days ago, young black bear cubs and their sows are now awakening earlier from hibernation each year because of warmer temperatures, and finding scarce food stuffs forcing them to raid human garbage bins. Thereby putting humans at risk as well as the animals since in most Colorado counties it's 'three strikes' then out as in euthanized.

In a second study by James Watson of the University of Queensland  the IUCN Red List has been referenced noting it only considers seven percent of mammals and four percent of birds as threatened by climate change and severe weather. The IUCN often drags behind the latest science – many species wait decades for an update while most species on Earth have never been evaluated. In a press release related to his work Watson noted that in worst-case scenarios, species are simply vanishing, adding: "There has been a massive under-reporting of these impacts."

A third study – in PLOS Biology – found that more than 450 plants and animals have undergone local extinctions due to climate change. Local extinction, as its name implies, doesn’t mean the species are gone for good, but that they have vanished from a portion of their range. For example, the barren ground shrew has seen its range constrict aggressively as its tundra home warms. The Mountain Lion here in the mountain West has likewise seen its range dramatically reduced as well as its numbers.

According to John Wiens, one of the study's co-authors,  based at the University of Arizona:

“If global warming continues, species that cannot change or move quickly enough may go globally extinct,”

It needs to be pointed out that such global extinctions have already happened. Last year, scientists discovered that the Bramble Cay melomys – an Australian rat-like rodent – went extinct  (it was last seen in 2007) due to rising seas inundating its tiny coral island.  It’s the first mammal confirmed to be pushed to extinction entirely due to climate change from our fossil fuel addiction.

Wiens’ study also found that local extinctions were happening more in the tropics than in temperate areas. This is worrisome since the tropics harbor the vast bulk of the world’s biodiversity, with many tropic species still unstudied and even undiscovered by scientists. But all the studies' authors have been alarmed by the fact changes are rippling even beyond single extinctions.

“We now have evidence that entire ecosystems, some the size of entire states within the USA, are changing in response to climate change,” said Scheffers. He pointed to kelp forests that he said “are dying” and being replaced by rocky, less-productive ecosystems.  Here in Colorado one in 14 trees is now dead and the number of gray-brown still standing-dead trees has increased 30 percent since 2010 to 834 million, the state’s annual survey has found. The dying trees are  largely the result of massive infestation by the mountain pine beetle -  now able to reproduce more rapidly because of warmer than average winters.

So the land forests are affected as well as kelp forests. The latter are made up of giant brown algae, with kelp as tall as trees providing essential nurseries for fish, protecting coastlines against worsening storm surges, and storing vast amounts of carbon.. But increasingly warming waters combined with ocean acidification is taking its toll.

UF's Scheffers expects more “ecosystem shifts,” as scientists describe them, in the future. Cloud forests are at risk of becoming high altitude grasslands, coral reefs of becoming algae-dominated ecosystems, and Arctic sea ice is being trans open ocean.  In the words of John Wiens:

“Given what we are seeing now, just imagine what will happen to all these species when temperatures increase by four of five times that amount,”

If  our global market and consumption society doesn’t reduce its fossil fuel addiction – and quickly – scientists estimate that temperatures could rise 4-5 degrees Celsius by the end of the century. Such a rise would be not so much catastrophic as apocalyptic. The difference is that you might barely recover from the first, though be without key resources (like rebuilding after being hit by an F5 tornado) but in the latter, there's no coming back, period.  Scheffers again:

“One thing that is certain is that this global response to climate change points to an increasingly unpredictable future for humans,” 

According to Scheffer’s research, humans will see a drop in productivity for various crops or timber species, a drastic loss in marine fisheries, a potential rise in new diseases (Avian flu?)  as well as diseases (like dengue fever)   spreading to places they’d never been before. Meanwhile, declines in coral reefs, kelp forests and mangroves will lead to more lives lost in climate-fueled storms. Loss of global biodiversity will also have knock-on effects in societies around the world, from less productive ecosystems to impacts we simply can’t predict today.

Scheffers said of his research in a press briefing:

“I was not surprised. But I was alarmed. The extent of impacts is vast and has impacted every ecosystem on the Earth.”

Scott Pruitt, his denier brigade and most of the mainstream media might well dismiss these warnings as "alarmist".  But it’s past time responsible people set off the alarms – screw Trump, Pruitt and the Trumpkins who inveigh against reality with their 'fake news' balderdash.  Indeed, as David Suzuki and others have noted the alarms should have started ringing in the 1980s and been deafening by the early 1990s.  Alas, the well- funded deniers went into full obstruction and distortion mode, often assisted by a complacent media more invested in "balanced" reporting than scientific reporting.

For his part, Scheffers believes there is still time to avert the worst, including:

“Governments investing and committing to reducing carbon emissions and protecting natural ecosystems that increase resilience to climate change not only for nature but for people as well,”

He added:. “These include large areas of connected forests which cool local and regional climate, pristine coral and oyster reefs that not only provide food but reduce storm surges, and well managed watersheds that will maintain adequate fresh water.”

Wiens agreed, but added that “there also needs to be more, bolder, large-scale

All of which is well and good, but ignores one huge, monumental problem: With Trump and his cabal in office none of those investments will be done to change the dynamic. Perhaps the largest CO2 polluting nation will also now be doing more coal burning, essentially cancelling out what any other nations do or don't do.

Without Trump and his gang in place there might have been a chance to achieve what Scheffers and Wiens suggest, but the odds with them there? Basically zero.  Even if by some miracle we managed to oust Trump and the Reepos, four years lost would be four years too many.

We are in for a rough climate ride and it's only beginning. Buckle up!

Stokes Parameters and Their Use In Radio Astronomy

In the ideal world of radio astronomy observations, one's choice source would likely be one for which the radiation is monochromatic or single frequency. In this case one deals with completely polarized waves and the associated constants (E1, E2,  δ ) are easily computed, with E1, E2 the E-field amplitudes. 

In the real world of radio astronomy, the objects detected display partial polarizations which must be further analyzed to reveal reliable information. Electromagnetic waves are  polarized when their E- field  components are preferentially oriented in a particular direction.

Other forms of polarization include:

Linearly or horizontally polarized: I.e. the E- vector is confined to one  (horizontal) plane

---------à E

Vertically polarized: I.e. the E- vector is confined to one  (vertical) plane

^ E

!
!
!

Circular: The E-vector rotates through 360 degrees

Elliptic: any polarization not circular or plane.

In addition to source polarization, the orientations of the sources in space relative to the observer must be taken into account using something called "the Poincare sphere".  This is where the Stokes parameters enter. Thus, emission from celestial radio sources within any finite bandwidth D u  consists of the superposition of a large number of statistically independent waves for a variety of polarizations. The resulting wave is then "randomly polarized" and the electric field components may be written:

E _x  = E1 (t) sin  (wt)

E _y  = E2 (t)  sin (wt  +   δ (t))

where all the time functions are taken as independent. Also, we have the time variations of E1(t), E2(t) and δ (t)  as slow compared to that of the mean frequency, i.e.  2 p u   , being of the order of the bandwidth  D u  .

The most general state is one for which the wave is partially polarized. Waves emitted from celestial sources are generally of the partially polarized form. In traditional radio astronomy, one deals with partial polarization by making use of the Stokes parameters, first introduced by Sir George Stokes in 1852. Reference may be made here to the attached diagram in order to fix ideas on how the parameters are developed and used.   We can write in the first instance:

E _x  = E1  sin (wt -    δ1)

E _y  = E1  sin (wt -    δ2)

where ( δ1 -   δ2 )  = the phase difference of  E _x   and  E _y  .

Expanding the two preceding equations and equating the  sin wt   and  cos wt   terms with results from two other equations, one obtains:

With each of the Stokes parameters, I, Q, U and V now quantitatively defined. Again, Z is the intrinsic impedance of the medium, while S _x  denotes the Poynting vector for the wave polarized in the x direction, and  S _y  denotes the Poynting vector for the wave polarized in the y- direction. It follows from this development that: 

I²  =    Q²  +  U ²   +   V²

And:

U/ Q  =    tan 2 t

And:

V/ S  =   sin  2 e   =    V/   Ö (Q²  +  U ² +  V²)

Note the above relations apply to a completely polarized wave.   Three specific cases are worth considering to fix ideas (AR is aspect ratio):

Case (1) Left Circularly Polarized Wave:

We have:  S _x   = S _y  ,   AR = 1, I = S, Q = U = 0,   V = S,  e   = 45 deg   

Case (2):  Right Circularly Polarized Wave:

I = S, Q = U = 0,   V = -S

Case (3)  Linearly Polarized Wave:

We have:   t    =   0,  S _x   = S,  S _y  =  0 ,  AR = oo,    e   = 0

I = S,  Q = S,  U = V = 0

For a completely unpolarized wave we have:  S _x   = S _y    and   E1   and  E2   are uncorrelated. The condition: Q = U = V = 0 is also a requirement.  Hence, a nonzero value for any of these quantities indicates a polarized component in the wave.  Also important is the degree of polarization, defined as the ratio of the completely polarized power to the total power or:

d  =  Ö (Q²  +  U ² +  V²) /  I

(For 0 < d < 1.)

It can be seen from the above, that d = 1 for a completely polarized waves and d = 0 for a completely unpolarized wave. (If two waves have identical Stokes parameters the waves are identical). It is also sometimes relevant to express a presence of an unpolarized component, i.e. if the Stokes parameters are expressed in both, say with appropriate subscripts u, p..  Thus one may find:

I =  S _u  +    S _p      =  S _u  +  S _xp   +   S _yp

Q  =       S _xp   -    S _yp

U   =     ( S _xp   +   S _yp )  tan 2 e  

V  =    ( S _xp   +   S _yp ) sec  2 t   tan 2 e  

  Normalizng the Stokes parameters entails dividing the time-averaged ([ ] ) equation:

I   =   [E1²] / Z  -   [E2² ] / Z

By: V =   2/ z {  E1 E2 sin δ ) = S sin 2 e 

Then we have for the normalized Stokes parameters:

s ₀  =   I / S   =   1  

s ₁  =   Q / S   =  (S _x   -  S _y) _/    S  =  [ cos 2t cos 2 e ]

s ₂  =   U / S   =  2/ Z  (E1 E2 cos δ  ) / S =  [ sin2t cos 2 e ]

s ₃  =   V  / S   =  2/ Z  (E1 E2 sin δ  ) / S =   [  sin2e  ]

Where 2 e  denotes latitude  and  2t  longitude on the Poincare sphere, while the new symbols ( s ₀  .....s ₃ ) are now introduced to represent the normalized Stokes parameters. The Stokes parameters can then be expressed as matrices of the column type, e.g.

S[  s _i ]  =

S [s ₀]
   [s ₁]
   [s ₂]
   [ s ₃]     i=0, 1, 2, 3

Where S is the total Poynting vector in W/ m²

And:   s ₀  =      1

s ₁  =   [E1²]  -   [E2²  ] /  ZS

s ₂  =   2 / ZS  [E1 E2 cos δ ]

s ₃  =   2 / ZS  [E1 E2 sin δ ]

If a partially polarized wave can be regarded as the sum of a completely polarized wave  and a completely unpolarized wave  we can write:  S[  s _i ]  =

S [s ₀]
   [s ₁]
   [s ₂]
   [ s ₃]

= S [1  -  d]    +
       [ 0]
       [0 ]
       [0 ]

S  [  d ]
    [ d  cos 2t cos 2 e ]
    [d  sin2t cos 2 e ]
    [d  sin2e  ]

where d is now the degree of polarization given in terms of the normalized Stokes parameters:

So, for a partially polarized wave:

s ₀    =   1

s ₁    =  d  cos 2t cos 2 e 

s ₂    =  d  sin2t cos 2 e

s ₃    =  d  sin2e

Note that circularly polarized wave components may also be used which will then rely on the quantity AR or the ratio of the major axis to the minor axis for the polarization ellipse (top diagram).. Then, e.g. for a completely polarized wave:

AR  =   (E _L  +  E _R ) /  (E _L  -   E _R )       =  cot e

Where the numerator and denominator shows left and right circularly polarized E-field components. If then, E _L  >  E _R     the AR is positive (left-handed polarization) while if  E _L  <  E _R  the AR is negative, or right handed polarization.  Then from the first AR equation we can see:

cos 2 e  =      2 E _L  E _R  / (E _L ² +  E _R ² )    =   (AR²    -  1) / (AR²   +  1)

Also:

sin 2 e  =  (E _L ² -   E _R ² ) /  (E _L ² +  E _R ² ) =    2AR/  (AR²   +  1)


There are many occasions for which the "coherency matrix" C will also be used. Here we are concerned with the response of the detection system, antenna  -receiver etc. and treat the detector as one for radio photons. (In regular optics the applicable matrices are called coherency matrices.

The coherency matrices and Stokes parameters for different types of radio wave are indicated below (the Stokes parameter will be given as a row matrix to save space but it is understood it must be rotated to a column form to be proper):

1) Unpolarized wave:

Stokes parameters:   [1..0..0..0]

Coherency matrix:  C =

½ [1…..0]

     [0…..1]

2)  Right circularly polarized wave:

Stokes parameters:  [1..0..0..-1]

Coherency matrix:  C =

½ [1…..-j]

    [j…..1]

3) Left circularly polarized wave:

Stokes parameters:  [1..0..0..1]

Coherency matrix: C =

½ [1…..j]

    [-j…..1]

4) Partially polarized wave (d = 1/3, or completely polarized wave plus part linearly polarized with   t =  45 deg)

Stokes parameters:  [1..0..1/3..0]

Coherency matrix:  C =

½ [1…..1/3]

     [1/3…..1]

Example Problem:

If a radio wave has characteristics d = 1 and  AR = 1, find the normalized Stokes parameters and the coherency matrix.

We have for the normalized Stokes components:

s ₀    =   1

s ₁    =  d  cos 2t cos 2 e 

s ₂    =  d  sin2t cos 2 e

s ₃    =  d  sin2e

And:   cos 2 e  =   (AR²    -  1) / (AR²   +  1)

=  (1  - 1)/ (1  + 1) =  0/ 1 = 0

Also:   sin 2 e  =  2AR/  (AR²   +  1)   =   2(1)/  (1 + 1) = 1

Then:

s ₀    =   1

s ₁    =  d  cos 2t cos 2 e  =   0

s ₂    =  d  sin2t cos 2 e  =   0

s ₃    =  d  sin2e    =  (1) (1) =  1

So;  Stokes parameters are:  [1..0..0..1]

For which coherency matrix: C =

½ [1…..j]

    [-j…..1]

Problems:

1) Write an expression  for the degree of polarization given in terms of the normalized Stokes parameters.

2)  Write a matrix equation for a completely unpolarized radio wave using a left circularly polarized wave and a right circularly polarized wave.

3) The coherency matrix of some individual radio wave is given by:

½ [1…..0]

     [0…..0]

Show how the  resultant unpolarized wave's coherency matrix may be obtained by showing the C-matrix for the other wave needed to combine with the individual wave above.

4) Four radio waves are detected and analyzed and found to have the characteristics shown below:

a) d = 0

b) d =  ½      AR = 4     and   t=  135 deg

c)  d =  ½      AR = 4     and   t=  - 135 deg

d)  d =  ½      AR = 4     and   t=  45 deg

Describe the polarization properties of each and the differences between them.

Find the normalized Stokes parameters and the coherency matrices for these waves.

GOP Congress Now Has Your Internet Privacy For Sale - What Can You DO?

Unlike in Europe, where governments respect their citizens, our current incarnation sees us mainly as chattel to do with as it pleases, unless our votes are needed. For example, in the EU strict privacy rules apply to protect citizen information. It cant just be bought and sold at the drop of a hat, nor can one - on doing a Google search for a name- find all kinds of options for buying further info, records etc., including address, voting record, bank account balance etc.

Contrast this with the GOP congress which on Tuesday last week voted to overturn privacy rules that previously required telecom companies to get customers' permission before sharing their web browsing and app history use with third parties. And, of course, Trump is also in favor of this - given he respects no citizens, even his own voters (mocking them not long after the election Nov. 8th by saying at a Louisiana rally, "Yeah folks, I just needed your votes, don't need you anymore!")

So no surprise that the White House announced Wednesday that Trump intends to sign the bill. This then would formally undo the rules adopted last fall by the Federal Communications Commission. It will be a boon to Verizon Communications, Inc., Comcast Corp. and A.T. and T. Inc. which are all building huge digital ad businesses.

What sort of information will now be up for grabs and sale to the telecoms? It includes: your full website browsing history - including porno sites, all the apps you use, the TV shows you watch, the stores and restaurants you patronize, the lists of all your Facebook friends, all the news articles they read (as well as you), where they bank and shop - along with their physical location. In other words, it presents a veritable windfall to any sharks wanting to track you or your circle - and target you all for future ads, pop-ups.

Basically, this disgusting congressional action, mainly on the part of the "business party" or GOP,  now gives companies and ad mongers comfort they can profile any customer behavior.  If you, for example, happen to visit some favorite porno site, your frequency of visits will now be tabulated and sent to any porn companies that may find that info useful. The next thing you know you will be receiving scads of porn catalogs in the mail, and likely in email too.

If you read regularly nutso conspiracy tripe from sites like InfoWars and Breitbart.com then those tracks will also be peddled to whoever wants them. Basically also, you won't know who gets that information or exactly how they use it.

Is there anything you can do about it? Probably the easiest tactic is to regularly go to "private window browsing". If you have Windows 10, you will be able to spot three tiny dots in the upper right corner of your screen. If you left click on it, you will see two upper options: new window, and new private window.  By selecting the second, you get a search field within which any entries you type will be private, in other words not accessible to the snoops.

What you shouldn't do, if  you value your privacy, is nothing.

Will Wonders Never Cease? Trump Tweets His Resignation!

Merriment abounded this morning as fake, unqualified Bozo Trump finally realized the magnitude of the job of President was too much for him, and tweeted his resignation. For any scoffers who believe this to be another April Fool joke, think again! The image above was pulled off 'Fox 'n Friends' this morning, photographed and scanned.

Why is Trump resigning? Clearly, because he finally realizes the job of President of the United States is far too big for him, too much to deal with. It's easier to just jet off to Mar-a-Lago every other week and now he wants to make that a permanent getaway.

For those for whom this is nonsense, keep in mind Trump is basically a skill-free inheritance baby with a virtually unbroken lifelong track record of incompetence and failure.  (Look at how this sleazoid just this week had to cough up big bucks for his Trump University scam.)    Having been used to easy real estate property flips and scams,  of course he'd find that running the U.S.  government  is a tougher job than lending his name to a bogus university, casinos and mail-order steak delivery scams run by other people.

But why would such an ordinary job loser also lose at being President when he conducted such an offbeat, high powered campaign? Because  'the Donald' is a world class, historically ignorant buffoon who couldn't tell the difference between his face and his ass even if they weren’t identical to each other. His followers, of course, were enamored of his performances on campaign stages - but only because they'd never beheld such magnitude of buffoonery before. Also, because many saw his performances on 'The Apprentice' and approved of the 'you're fired!' paradigm, they believed he'd use to good measure in "draining the swamp" in Washington. Not realizing this was another bait and switch and Donny boy actually wanted to fill the swamp.

Yes, being the president was a harder job than Trump expected, 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.  “Overseeing a family business” was great because Trump could call all the shots and all he had to do is yell 'Jump!' and lackeys would ask 'How high?'

But the presidency isn't a dictatorship nor can a president act like a CEO. Which is actually a more generous way of saying that Trump's current position isn't like his last gig as the ornamental figurehead of a penny-ante hustling operation run by his hare-brained children.  Who even in their vacuity knew better than to let him handle any responsibility more sophisticated than ogling the Miss Universe contestants.

So why - other than his daft, myopic supporters- would anyone be surprised now that his administration is swirling slowly down the drain?  Doing things that one is not qualified to do is tough! Only a dolt, blind man or fop could have failed to predict that the presidency would be a challenge for an addle-pated, butter-soft septuagenarian - who never served in the military, or in any capacity of the government.

Trump's aversion to real work quickly became apparent as he mired himself in tweets, watching cable TV (mainly FOX) and signing phony Executive orders. Trump's investment of time for these distractions was originally 5 hours daily and quickly expanded to 9 to 10 hours, but he found even this wasn't enough to distract from responsibilities. Hence, opting to go to Mar-a-Lago more and more, to be able to host his little rich country club boys at posh dinners or play golf with them.... all on the taxpayer's dime.

Trump then, has finally decided that it's better to be a full time wealthy playboy than a president - where people actually expect real results. Not prepared or qualified to deliver results, his only choice has been to back out.

Besides, better to resign now than wait for all the shoes to drop with the oncoming investigations into his Russia connections.

Good riddance!

See also:

http://www.latimes.com/projects/la-ed-our-dishonest-president/