Solutions to GRE Physics Problems (2)

16) This solution can be treated classically since 6 x 10⁶ m/s is much less than c, the speed of light. Thus:

dv/dt = a and upon integration:

v - v(o) = at

and dx/dt = v

Integrating a second time one obtains:

x - x(o) = v(o)t + ½ at²

Now: a = -1.2 x 10¹⁴ m/s/s

v(o) = 6 x 10⁶  m/s and the final velocity is: v(f) = 0

So: v(f) - v(o) = at = 0 - 6 x 10⁶ m/s = (-1.2 x 10¹⁴ m/s/s) t

whence: t = [0 - 6 x 10⁶ m/s ] / (-1.2 x 10¹⁴ m/s/s)

t = 5 x 10^-8 sec

Now: x(f) - x(o) = v(o)t + ½ at²

= (6 x 10⁶ m/s)(5 x 10^-8 sec) - ½((-1.2 x 10¹⁴ m/s/s)(5 x 10^-8 sec)^2

x(f) - x(o) = 0.15 m = 15 cm

17) By Newton's law of universal gravitation, the force between the two masses is:

GMm /r²

From the integration of the above the potential energy can be obtained as: -(GMm/R) and the kinetic energy as KE = GMm/2R.

let d = 2R then the total KE available is:

GMm/d

Now: Momentum must also be conserved so (assuming conservation of linear momentum along the line of action):

mv(m) = M v(M)

But: KE = ½ [mv(m)² + M v(M)²] = GMm/d

Using algebra:

v(M)²[1 + M/m] = 2GM/d and

v(m)² + v(M)² + (m/M)v(m)² + (M/m)v(M)²= 2G/d{M + m}

v(m)² + v(M)² = 2G/d (M + m)

Therefore: v _R  =  [2G/d (M + m)]^`1/2

18) We define current as the rate at which charge flows, thus:


I = dQ/ dt so dQ = I dt, or Q = It

since I = 5 A (amps) = 5 coulombs/sec

then for 4 minutes (= 4 x 60 sec = 240 secs):

Q = 5 coulombs/sec (240 sec) = 1200 coulombs


19) The effective value of a sinusoidal emf is its root mean square value, defined:

v(eff)² = v(rms)² = ∫  [v(max)² sin wt dt] T = ½ v(max)²

(integrated from 0 to t)

so v(eff) = [½ v(max)²]^1/2 = 1/ [2]^1/2 (v(max)) or

v(eff) = Ö 2  (v(max)) / 2


20) Potential difference is the work needed to move a test charge (q) from point A to B, or:

V = W/q

In this case, W = 2 J and q = 20 C, so:

V = 2J/ 20 C = 0.1 Volt


21) For a parallel plate capacitor,

C = k k(o) A/ d

where k(o) = 1 for air, and k is the relative dielectric constant: or

k = k(mica) k(o) = 6 k(o) = 6

Then C(mica) = 6 (15 m mF) = 90 m m F

(A/d is the same in each case)


22) The charge on the electron, q = 1.602 x 10^-19 C

Again, from Problem #18, the current is charge moved per unit time:

I = dQ/ dt

In this case, since the angular frequency is the key information given, and it is related to time via:

 w = 1/T

then:

I = q w = (1.602 x 10^-19 C) (6.6 x 10¹⁵ /sec)

I = 1.05 x 10^-3 amp, or I = 0.00105 A


23) For a plane mirror the applicable law of geometrical optics is that the image appears to be as far behind the mirror as the object is in front. Therefore, one's eyes must be focused to a distance of 40 cm.

24) We apply the lens makers' formula here, see e.g. http://brane-space.blogspot.com/2011/06/introduction-to-basic-physics-thin.html:

1/ f = (n - 1) [1/r1 - 1/r2]

where f is the focal length of the convex lens (to be found)

n = 1.65 (the refractive index) and r1 = 40 cm

then r2 = -40 cm (recall for convex lenses, r2 = -r1)

Then:

1/f = (1.65 - 1) [1/40 cm - (1/ -40cm)]

1/f = 0.65/ 20 cm

So: f = 20 cm/ 0.65 = 31 cm


25) In this nuclear reaction, the charge and nucleons must be conserved, meaning that the upper index total on the right side must be equal the upper index total on the left, plus the lower number on the right must be equal to the sum of lower numbers on the left. The only one which fits this is (D)
8 O 17.

26) Here, we use the relativistic form for the kinetic energy:

K= Mc²    -  M(o)c²

We are given the rest mass is 106 MeV so M(o)c² = 106 MeV

We also have: K = 4 MeV, then:

Mc² = K + M(o)c² = 4 MeV + 106 MeV = 110 MeV

We also know:

M = M(o)/ [1 - (v/c)²)]^1/2

But K = M(o)c² [1/ (1 - (v/c)²)]^1/2) - 1]

Therefore (with some algebraic working):

(v/c) = 0.28

or

v = 0.28c = 84,000 km/sec

27) The system depicted in the graphic below  is the one used for solution here.



The Lagrangian is the difference between kinetic and potential energies:

L = T - V

And from the system:

T = ½ m1v1² + ½m2v2² = ½ (m1 + m2)x' ²

V = (m1 x - m2 x) g

So:

L = ½ (m1 + m2)x' ² - (m1 x - m2 x)g

= ½ (m1 + m2)x' ² + (m2 x - m1 x)g


28) Applying force balance:

m2 g - T = m2 a

and:

m1 g - T = -m1a

(where T is the tension in the string.)

so: Using algebra to solve the simultaneous equation:


m1 m2 g - m1 T = m1 m2 a

m1m2 g - m2T = -m1m2 a
----------------------------

2 m1m2g - (m1 + m2) T = 0


and T = (2 m1m2 g) / (m1 + m2)

29) The efficiency of a Carnot cycle engine is:

e = 1 - Q(R) / Q(W)

where Q(R) denotes the heat returned to a reservoir at T2, and Q(W) is the heat withdrawn from a reservoir at T1.

Then:

Q(R)/ Q(W) = T2 / T1

and

e = 1 - Q(R) / Q(W) = 1 - (T2/T1)


30) From the quantum principle for the total angular momentum:

L = L1 + L2........[L1 - L2]

Thus:

L = 5, 4, 3, 2, 1

The Latest NSA Outrage - And What Snowden Sought to Protect

Even as another militarist (Michael Rogers) is named by Obama to head the NSA, we've learned about the latest NSA outrage: that NSA and its Brit sidekick GCHQ are grabbing up data from smart phone apps, and even games like 'Angry Birds'. Enjoy  'Angry Birds?  Then you're likely now compiled into the NSA's "Main Core" program, along with proven "enemies of the state" and others (e.g. fracking protestors, and OWS folks) deemed such.

We know that with each new generation of mobile phone technology, ever greater amounts of personal data pour onto networks where spies like those ensconced in NSA can pick it up.   According to dozens of previously undisclosed classified documents, as reported by the NY Times yesterday, among the most valuable of those unintended intelligence tools are so-called leaky apps that spew everything from users’ smartphone identification codes to where they have been that day.

As The Times observes:

The N.S.A. and Britain’s Government Communications Headquarters were working together on how to collect and store data from dozens of smartphone apps by 2007, according to the documents, provided by Edward J. Snowden, the former N.S.A. contractor. Since then, the agencies have traded recipes for grabbing location and planning data when a target uses Google Maps, and for vacuuming up address books, buddy lists, phone logs and the geographic data embedded in photos when someone sends a post to the mobile versions of Facebook, Flickr, LinkedIn, Twitter and other services.

Of course, WH tool Jay Carney denied it all, but what would you expect when the snoops probably have all sorts of stuff on him, as well the Obama WH - likely making subtle threats to undermine Obama's legacy and programs (the ACA)  if he doesn't "play ball".  Bear in mind the snoops also had JFK under the gun (literally), as they laid out an assassination program against Fidel (ZR/Rifle) which was then turned on Kennedy. As Peter Dale Scott has reported (Deep Politics Quarterly, Jan. 1994)  this was operated under the NSA and the CIA's  William Harvey with the 'Staff D' connection betrayed on the front cover of Oswald's 201 file. (This indicated a SIGINT or signals intelligence operation run in concert with the National Security Agency or NSA.)

The Times went on to note:

The eavesdroppers’ pursuit of mobile networks has been outlined in earlier reports, but the secret documents, shared by The New York Times, The Guardian and ProPublica, offer far more details of their ambitions for smartphones and the apps that run on them. The efforts were part of an initiative called “the mobile surge,” according to a 2011 British document, an analogy to the troop surges in Iraq and Afghanistan. One N.S.A. analyst’s enthusiasm was evident in the breathless title — “Golden Nugget!” — given to one slide for a top-secret 2010 talk describing iPhones and Android phones as rich resources, one document notes.

Can anyone believe this? And also the arrogance to outright deny it when the evidence is clear?

The Times again:

Two top-secret flow charts produced by the British agency in 2012 show incoming streams of information skimmed from smartphone traffic by the Americans and the British. The streams are divided into “traditional telephony” — metadata — and others marked “social apps,” “geo apps,” “http linking,” webmail, MMS and traffic associated with mobile ads, among others. (MMS refers to the mobile system for sending pictures and other multimedia, and http is the protocol for linking to websites.)

You want to know why the congress can't support food stamps for our millions of hungry kids, or unemployment benefits for  out of luck long term unemployed? Look no further than this nugget from the Times:

As the program accelerated, the N.S.A. nearly quadrupled its budget in a single year, to $767 million in 2007 from $204 million, according to a top-secret Canadian analysis written around the same time.
 

Got that? Over three fourths of a billion for these snoops. For what? Well to keep the national security state expanding! And they know millions are dumb enough to believe the transparent bilge of "protecting us from terrorists" when the real terrorists are the random mass killers that can open up on innocent kids in a school, a mall, or a movie theater. But the NSA doesn't want to face down the likes of the NRA, oh no! Nor does a cowardly congress. It's easier to confect bogeyman terrorists than to challenge the real enablers. As Baltimore Sun columnist Dan Rodricks wrote after the mass shooting in the Columbia Mall:
 

"We have spent billions of dollars in the fight against terrorism — the National Security Agency is maybe 10 miles from the Columbia Mall — when the real terror is right here in our midst, accounting for thousands upon thousands of premature deaths by homicide and suicide, most often impulsive acts made possible by access to guns."

 The Times went on to point out:

Smartphones almost seem to make things too easy. Functioning as phones — making calls and sending texts — and as computers — surfing the web and sending emails — they generate and also rely on data. One secret report shows that just by updating Android software, a user sent nearly 500 lines of data about the phone’s history and use onto the network.
 

Meanwhile, there remain millions of pretend Americans - those who don't deserve liberty because they overvalue a phony security - who will blow these revelations off and still call for Edward Snowden's head. (That he be "brought to trial" - when the format would not even allow him to call public witnesses on his behalf).  I have nothing but contempt for these Americans when my own ancestor, Conrad Brumbaugh, fought in the Revolutionary War for the liberties now being trampled under.

So what did Snowden seek to protect by his disclosures? None other than privacy! As I noted in my June 8 blog post from 2013, secure in one’s person, house, papers, effects” the Fourth amendment implies PRIVACY! These are after all MY private papers, my private effects, my house, etc. If an inherent right to privacy was a myth then by all accounts being secure in one’s person, papers, effects wouldn’t matter.  Beyond that, privacy encapsulate an affirmation of our worth as autonomous humans as opposed to being slaves and pawns of the state.

The cornerstones can be said to be:

-  Personal autonomy (the ability to make personal decisions)

- Individuality  
-  Respect
- Dignity

 -Worth as human beings
 

As one privacy source puts it:

Privacy allows us to make our own decisions free from coercion, to totally be oneself and potentially engage in behavior that might deviate from social norms. It allows us the time and space for self-evaluation. Informational privacy is seen as enhancing individual autonomy by allowing individuals control over who may access different parts of their personal information. It also allows people to maintain their dignity, to keep some aspect of their life or behavior to themselves simply because it would be embarrassing for other people to know about it.

Privacy also allows people to protect their assets or to avoid sharing information with others who would use it against them, such as discrimination by employers, educators, or insurers. The ability to control one’s information has value even in the absence of any shameful or embarrassing or other tangibly harmful circumstances.

Privacy is also required for developing interpersonal relationships with others. While some emphasize the need for privacy to establish intimate relationships, others take a broader view of privacy as being necessary to maintain a variety of social relationships.

By giving us the ability to control who knows what about us and who has access to us, privacy allows us to alter our behavior with different people so that we may maintain and control our various social relationships. For example, people may share different information with their boss than they would with their doctor, as appropriate with their different relationships.

Most discussions on the value of privacy focus on its importance to the individual. Privacy can be seen, however, as also having value to society as a whole. Privacy furthers the existence of a free society. Large databases, potential national identifiers and wide-scale surveillance, can be seen as threatening not only individual rights or interests but also the nature of our society. For example, preserving privacy from wide-spread surveillance can be seen as protecting not only the individual’s private sphere, but also society as a whole: privacy contributes to the maintenance of the type of society in which we want to live.

------

Minus privacy and its inherent defense via the 4th amendment, we thereby become like  "Winston" the pathetic character from Orwell's '1984' whose flat is riddled with two-way monitors with the image of 'Big Brother'. Winston can't even take a crap without Big Brother's beady, baleful eyes staring down at him. THINK about THAT! When he has sex Big Brother is in the room, and the first thing in the morning Winston's ordered by the Oceania Officer on the screen to do his calisthenics and if he loses tempo for an instant he must repeat them.

This is the world we effectively have now via NSA mass snoop technology, and the only reason over three fifths of my countrymen want Snowden taken to a trial is they can't see the value of what he's done. They can't see that had he not done it there'd have been NO scrutiny at all on this secretive agency which Sen. Frank Church once warned us about in frightening terms:

 "I know the capacity that is there to make tyranny total in America. And we must see to it that this agency and all agencies that possess this technology operate within the law and under proper supervision so that we never cross that abyss. That is the abyss from which there is no return"- 

 

To me, this makes the mock Americans who want Snowden tried no better than the good Germans of the Nazi era.  Those degenerates sided with Hitler, praised him and embraced his ideology while turning a blind eye to the multifold wrongs. The last straw was the Enabling Act (1933) which essentially obliterated the last vestiges of the Weimar Republic. These "good Germans" - state loyalists to a fault-  lost the last vestiges of their democracy from right under their miserable noses because they were too dismissive of the clear warning signs all around them. They had it nice and comfy thanks to Herr Hitler, so why rock the boat?

After our military dragged these “good Germans”  into the remnant concentration camps-  to see first hand what their Fuhrer and his minions did- they wished to hell they had rocked the boat. They were marched in and forced to look at the naked, gassed bodies stacked like cordwood until they puked.

Meanwhile, our then military leaders rounded up the remaining high profile Nazis, including Hermann Goering (see photo below) ,


 and dragged them before a Tribunal to be accountable for their actions. "Following orders" was not an acceptable defense! Under the Nuremberg laws and specifically its Principle IV,  the defendants had to have operated under a higher moral order or conscience, damned the "orders", "oaths" or any other contrived legalistic mumbo-jumbo designed to maintain silence in the face of atrocity.

If a person dismisses the NSA's latest outrages, namely the smart phone metadata seizures, or doesn't believe privacy ought to exist anymore anyway because "someone can always find out about you" then whether they know it or not they're like the good Germans. I'd even go further, and say they're no different from Martin Borman or Hermann Goering.
 

What would my Revolutionary War ancestor  think about these "Americans" now,  nearly 240 years after the War of Independence? He’d likely barf nonstop at the spectacle of what too many of the current crop have devolved to: arrant, ignorant, whiny, weak, sissy consumers,  who’d rather calumniate someone trying to preserve their liberties than recognize the real threat to their existence. He’d then likely crawl back into his grave shaking his head but echoing Benjamin Franklin’s famous quote:

"Those who would sacrifice liberty for a temporary safety deserve neither liberty nor safety"

 Am I a civil liberties “extremist”? Damned right if that means a citizen who understands that the Constitution is not just “a piece of paper” (as Bush once called it) and that the rights inherent in the Bill of Rights are real, apply to individuals, and not mere “compromise abstractions” but rather hold the key to American identity – what truly sets us apart. And once those rights are gone, believe me they won’t be coming back!

Those quaking in their boots at Snowden's continued disclosures would do well to bear in mind the words of Bruce Schneier, a security specialist, who wrote in The New York Times last July 3rd:


"The argument that exposing these documents helps the terrorists doesn't even pass the laugh test; there's nothing here that changes anything any potential terrorist would do or not do."

BINGO!

See also:

http://www.smirkingchimp.com/thread/robert-scheer/53895/who-needs-the-gestapo-when-you-have-angry-birds

 

 

 

 

Tom Perkins: Another Whiny, Rich Capitalist With His Head Up His Ass

It would seem that a normally functioning rich (upper 1/2 of 1/2 of 1/2 percent)  guy, in the wake of how these assholes helped cause the 2008 financial meltdown, would shut the fuck up and certainly keep a low profile. But no, it appears some are too brain dead to know that and enter now Tom Perkins  - a billionaire venture capitalist-  pontificating in a letter to The Wall Street Journal in which he wrote of 'parallels' between the treatment of Jews in Nazi Germany and what he describes as the "progressive war on the American one percent".

Can anyone seriously believe this?  Does this elitist jerk off even know about the treatment of the Jews in Nazi Germany? Well, if he did, he wouldn't even remotely think of making such an insane analogy. And yet we know the Right does this all the time, whether out of reaching for the most outrageous comparisons they can find to tear down those who hold them up to moral scrutiny, or because they themselves know it's a lie - but figure there are enough dunderheads out there (who lack an informed knowledge of history) to accept the analogy. (Look no further than the 61% of Americans who actually believe Edward Snowden ought to be brought to trial.)

Let me deliver an image for this rich rat fucker and his entitled Neolib, market exploiting pals to show how the Jews were treated in Nazi Germany. It is an image of dead, gassed bodies piled up outside of the concentration camp known as Mauthausen.    The photograph was made by Russian soldiers - who came upon the grisly scene- in May of 1945.   

Perkins and his pallies may wish to know WHO was responsible for this ghastly scene, emblematic of the annihilation of 6 million people. He may be amazed, because he's obviously ignorant, that it in large measure was facilitated by rich business men and venture capitalists like himself!

Ian Kershaw in his masterpiece ‘Hitler Nemesis’, observes,  p. xxxiii):

“Leaders of big business, though often harboring private concerns about current difficulties and looming future problems for the economy, for their part were grateful to Hitler for the destruction of the left-wing parties and trade unions. They were again ‘masters in the house’ in their dealings with their work force”

This was in reference to Hitler dispatching his S.A. thugs to beat and kill Socialists (the REAL ones, not the fake named "National Socialists") as well as communists and labor leaders. This was in addition to his reign of terror against Jews which saw them first labeled using yellow stars (sewn on to their clothing) with the word 'Jude', then deprived of their businesses (having them seized), then their homes - as they were dispatched to the concentration camps. (Buchenwald, Auschwitz, Treblinka, Mauthausen, etc.)

THIS is why the German capitalists, such as Krupp, Volkswagen, I.G. Farben (infamous for later manufacturing the Zyklon B gas used in the gas chambers) etc. threw their lot behind him and provided the political ballast, without which he could never have become 'Der Fuehrer'.  They even helped design the gas chambers for maximum efficiency.

Of course, the rich, entitled swine like Perkins don't wish to hear this, but that is history. Anyone can check it out if they have a mind too, and aren't brainwashed by the Neoliberal capitalist corporate media imps.   As Dieter,a  former Wehrmacht soldier put it, when I met him and two other former Wehrmacht in May, 1985:

"It was disgusting how far the big businessmen went to support Hitler! Had it not been for them I doubt he'd ever have been Chancellor or even gotten 15% of the vote in the 1932 elections.

Incredibly, Perkins was educated at the Massachusetts Institute of Technology and Harvard, disclosing that even a degree from the vaunted Ivy League can turn out worthless if one lets his mind go to seed.  Money --- well, all the money in the world is useless if one's psychological foundation is undermined by untruth, PR and bollocks. 

The dope concludes his letter by warning of a "very dangerous drift in our American thinking," before adding to that:  "Kristallnacht was unthinkable in 1930; is its descendent 'progressive' radicalism unthinkable now?"  (Kristallnacht, also known as the Night of Broken Glass, saw a series of coordinated attacks against Jews throughout Nazi Germany in November 1938.The attacks were carried out by Sturmabteilung  (S.A.)forces and non-Jewish civilians and were ignored by the German authorities. 91 Jews were killed and around 30,000 were arrested and incarcerated in concentration camps.)

Again, the leap to comparing Kristallnacht administered by Nazi thugs to what progressives might to do Perkins and his rich elites is so absurd as to be barely worth dignifying by response. One need only consider here how Perkins' security state allies came down on Occupy Wall Street back in 2011, even to the extent of aiming rifles at them in Houston! See e.g.http://www.thiscantbehappening.net/node/1494

This alone shows who holds the reins of power in this society, and it ain't OWS or any of its progressive allies! The notion that those (like OWS)  who've been smacked down by the U.S. national security state can just rise up under some powerful liberal "Fuehrer" to take out Perkins and his entitled elites is so laughable and pathetic as to rival the worst of the 'Dumb and Dumber' movies. In fact, I'd rate it as a qualifier to put anyone who opines it in an institution with thorazine and ECT administered thrice daily.


Others who've been exposed to Perkins' codswallop have described the businessman as a "rich idiot" and condemned the WSJ for printing the letter.  (Given the other tripe and trash they've published over the last few years, especially since Murdoch gained control, I'm not surprised at anything that appears in it anymore.)

New York Times writer Steven Greenhouse tweeted: 'As someone who lost numerous relatives to the Nazi gas chambers, I find statements like this revolting & inexplicable.'

They're "explicable" all right. They are coming from a rat who realizes most people are on to him and his entitled little rich fucks, bathing in their $65,000 copper tubs e.g.

And being able to hop off to St. Kitts every weekend for 18 holes while their wives get rose wine wraps as they gobble their $25,000 apiece Frrrozen Haute Chocolates  - - as millions of American kids starve because Rich-protecting Reepos have cut food stamps and unemployment benefits.

Oh, and why those cuts? So the Reepos can lighten the tax load on the rich entitled elites and gob-smacked global capitalists and hedge funders, and also cushion the military from having any significant cuts.


Let's understand, as Ian Kershaw and Konrad Heiden ('Hitler') document, that Hitler saved his most ferocious attacks for German Marxists, Communists and Socialists - who he automatically consigned to "racial traitors, non-German, non-Aryan". (In most of Hitler's writings he asserts Marxists are Jews based on the fact Karl Marx, author of Das Kapital, was a Jew.)   In one of his worst tirades, a radio speech on October 15, 1933 (from Konrad Heiden's book) Hitler shouted:

These ruinous and inferior characters have succeeded in arousing a world psychosis

This was tied to those he perceived as "Jews" in the American press, working in league with communists and Socialists in Europe to spread rumors about Hitler and demonize him. What happened is that the Jews - like ALL Socialists, Communists, and Marxists- were dispossessed of their worldly goods and bank accounts. By 1936 virtually no German Jew (just as no German Socialist, or Marxist) could own a business, or even own a personal bank account. Worse, if sighted on the streets they risked terrible beatings by gangs of S.S. thugs. Communists were often killed on sight.

If I have any fear-  any at all in this matter - it's that Perkins and HIS lot will invest hundreds of millions in the construction of mammoth concentration camps somewhere out of sight, and then pile PROGRESSIVES who criticize the rich into them, herd them into giant gas chambers  - and eliminate them with Sarin.

Maybe Perkins in his mind knows that's his ultimate plan, so wants to fob the paranoia onto the Left to disarm the stupid people in our midst, of which there are many.

More Complex Integrals

We continue now looking at more examples of complex integration and application of the residue theorem.

Example 1:

∫ _-¥  ^¥  (1 + x² )  dx / 1 + x⁴ 

Find the residues at exp(pi/4)  and exp(3pi/4)  and hence integrate the preceding:

Solution:

 Begin by identifying the singular points where:

f(z) = 1 + x²  / 1 + x⁴ 

Now:  We note that  1 + x⁴     =   0 when   x = (-1) ^¼   

Or the fourth root of (-1). The modulus of (-1) = 1 and the argument is p.  Therefore, by de Moivre’s theorem:

(-1  + 0i) ^¼     =   ⁴Ö(-1) [ cos (p + 2kp/ 4) + i sin (p + 2kp/ 4)]

k=   0, 1, 2, 3…….

At k= 0:     (-1) ^¼      =  cis (p/4)   =   exp (pi/4)

At k=1:   (-1) ^¼      =  cis (3p/4)    =      exp (3pi/4) 

(We’re  not concerned with k=2, 3……Why?)

We therefore take the semi-circular contour as shown in the graphic. So that:

ò_CR   f(z) dz  +  ∫ _-R  ^R  f(x) dx  =  2 pi  (sum of residues inside C)

As R increases without bound we have:

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


2 pi [Res f(z) at exp (pi/4) and exp (3pi/4) ]

For the case f(z) = p(z)/ q(z)   Then:

Res f(z) _{z = a}      =  lim _{z ® a}   [p(z)/ q’(z)]

Then for z = exp (pi/4):

Res f(z) _{z = exp pi/4}       =   lim _{z ® exp pi/4}      1 + z²  / 4 z³     =

1 +  exp(pi/2) / 4 exp(3pi/4) =  1 + cis (p/2) / 4 cis (3p/4)

Res f(z) _{z = exp pi/4}       =    1 + i /  4(- Ö2/ 2   + iÖ2/ 2   )

=  1/ 2Ö2 (i+ 1/ i- 1)

Rationalize the denominator to get:  2i/ -2  =  - Ö2i/ 4

Then for z = exp (3pi/4):

Res f(z) _{z = exp 3pi/4}       =   lim _{z ® exp 3pi/4}      1 + z²  / 4 z³     =

1 +  exp(3pi/4) / 4 exp(9pi/4) = 

¼ [exp(-3pi/4)  +  exp(- 9pi/4)] = 


¼ [ cis (-3pi/4)  + cis (- 9pi/4)]

Therefore:

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


2 pi [Res f(z) at exp (pi/4) and exp (3pi/4) }

=  2 pi {- Ö2i/ 4   +  ¼ [ cis (-3pi/4)  + cis (- 9pi/4)]}

Problem for Serious Math Mavens:

Obtain the integral for:

 ∫ _-¥  ^¥    x  dx / (x²   - 2x + 2)

 (Hint: Remember  the residue formula (i.e. for a given pole of order m) and your complex numbers!)

Can We Hope to Educate Fundies on the Different Types of Atheist?

In approaching the fundamentalist brain, one thing one must perpetually keep in mind is its inherent simplicity and the incessant demand for simplicity. Thus, rather than acknowledging the Big Bang as the source for inception of the cosmos the fundie will invent 'young Earth' and 'young Sun' nonsense because it's simpler. After all, you needn't learn about the physics of the Big Bang including the significance of the 2.7 K isotropic microwave radiation  -- you need only invoke the fairy tale of 'Adam & Eve'.

In a similar vein, in approaching evolution the dumb, semi-educated fundie will usually cherry pick one aspect to try to make hay out of (like the 98% chimp-human sharing of DNA) but entirely ignore the more difficult aspects such as that: 1) chimps and humans share the exact same cytochrome-c protein sequence despite 10^93 different possibilities for variation, and 2) the fact that the ape 2p and 2q chromosomes underwent telomeric fusion to become the '2' chromosome in humans. 

But this is what we've come to expect from  a lot that still believes ancient fables such as that a man could survive 3 days in a whale's belly (or 'great fish') despite it being full of hydrochloric acid, or that an 'Ark' could have been constructed to carry pairs of all the critters then on Earth including dinosaurs.

So it's not a surprise to learn that most also lack the intellectual capital or ability to appreciate the distinctions between atheists.  After all, it's simpler just to assume ALL atheists are of one type, e.g. "militant" if they even write something negatively about Xtianity.  Their pea brains fail to understand that writing something negative, say in a blog, is NOT the same thing as demanding the extirpation of religion - even fundy-ism.

Nor do these clowns grasp the distinction between agnostics and atheists. So we are obliged to help them out, and try to express some sympathy too.

Author George Smith[1] notes that the most vernacular usage confuses the meaning of agnostic, usually asserting it is “somewhere between belief and outright unbelief.” The way it is usually framed is that the agnostic “isn’t sure whether or not God exists.”  In fact this is an erroneous interpretation of the term.


What it actually, technically means is someone who asserts there is not yet adequate evidence to make a decision if one were possible. In other words, the issue must remain open-ended and unresolved until such time evidence appears. Thus, true agnosticism is not some dithering or fence-sitting between belief and unbelief, but rather acknowledgement of the impossibility of knowledge as it pertains to any ability to address the question in the first place.

But mayhap this is far too complex for the simplistic gray matter of fundies.

Smith himself (op. cit., p. 9) splits and parses the agnostic position even further, according to whether one is an agnostic atheist or an agnostic theist. The first avers that all supernaturalist propositions or claims are inherently unknowable (because they are untestable) by the human mind, and so not worth discussing any more than pink elves, unicorns or fairies. In this sense, unbelief isn’t really even necessary, since who would actually believe in a pink elf or unicorn? Well, then there’d be even less merit to arguing over an invisible Being! Thus, without any capacity for knowledge about it, it’s redundant.


The theist agnostic meanwhile, maintains a firm belief in the existence of God, but asserts the precise nature of this God is unknowable. On this basis, no attributes are assigned to try to define God’s nature.  We cannot say that any divine nature is “eternal”, or “omnipotent” or “omnipresent” because one simply doesn’t know. Thus, it makes no sense to use words to try to describe a deity. Belief is as far as one can go.

Related to definitions of atheists is the ability to distinguish between implicit and explicit atheists. In many ways, the implicit atheist is similar to the agnostic atheist except for the fact the kmowability or otherwise of supernaturalist propositions is not gone into. Thus, the implicit atheist simply avers to be withholding belief in any kind of deity already on offer by the theist. (The explicit atheist, by contrast, rejects it outright.)

As I noted in an article published in the Mensa Bulletin, March 1994:

"Let's be clear about what constitutes Atheism and what doesn't. The Atheist - to put it succinctly, absolutely withholds investing intellectual/emotional resources in any supernatural claim. Indeed the word Atheism itself embodies this definition: a-theos, or without god.”

What is happening here is not active disbelief, i.e. in making a statement “There is no god,” but rather simply passively withholding belief in a statement already made. Hence, the deity believer has made the positive claim. The ontological   (implicit) atheist’s  response is simply an absence of belief in it. No more and no less. It does not and never has implied  active disbelief, aggressive rancor or a vehement and militant opposition to the beliefs.


Let me quickly add here that this withholding of belief is the more natural position, as opposed to advocating belief, which is unnatural. Consider a different context: a neighbor runs over and informs me that aliens have landed in his yard in a spacecraft. Until I actually go over and try to verify his claim I am under no obligation to accept it as a statement of fact. Thus, the default intellectual position is always skepticism, irrespective of the claim made. This again is because the onus is always on the claimant to make good, not the skeptic to “disprove” it.

It isn’t difficult to see from the above context that the more conservative (and reasonable!) position is withholding belief until a claim is validated. One does not, after all, accept a claim then do further research. One ab initio doubts the claim and then sets out to devise tests to ascertain the validity!  And in the case of extraordinary claims (which certainly include “God” and visiting aliens in spaceships), “extraordinary claims demand extraordinary evidence” as the late Carl Sagan used to emphasize.

 

It also isn’t difficult to see that this is exactly analogous to the atheist withholding belief in a deity. After all, If God genuinely exists, why is he/she/it not uniformly perceptible,  at least in basic features, to all peoples? As we saw in previous blog posts (from 2011, mainly), there are nearly as many different versions of deity as there are people. (Or at least, religions!)

Another error made by the wayward fundie is assuming that a negative belief, i.e. that there is no God, is the same as being a positive belief, 'there is a God'.  An example of this would be what Lisa Kennedy spouted on a Bill Maher Real Time show in January, 2012, i.e.  if an atheist simply doesn’t believe in the supernatural or God he is expressing a belief, albeit a negative one. If expressing a negative belief, then it's the same as a positive belief such as invoked in religions, ergo he is professing a religion. This is utter nonsense.


It would be akin to asserting that if I decline belief in ghosts I still have “negative ghost belief” and therefore am a Ghostologist ! The error inheres in asserting that an absence of belief is the same as a belief. This error repeats the canard that the onus is on the non-believer to disprove the believer’s claim, instead of acknowledging it is impossible to prove a negative. Some sophists attempt to get around this by claiming it  "isn’t impossible to prove a negative". They argue that they can prove “there are no black balls” in a box by simply emptying the box out and finding all white balls. But this misrepresents the example, since the possibility of “all white” balls was never in question! The analogy is specious because the existence of God IS in question, and isn’t objectively verifiable, unlike counting real balls. The analogy trivializes the deity existence question while not validating the sophist.

In the end, it may be that because the brains of fundies are unable to perform ACH thinking - or what James Cheyne  has called abstract, categorical and hypothetical thinking (such as integrated into standard I. Q. tests, e.g.  the Raven's and Wechsler Similarities tests) then they cannot also be expected to discriminate between subtle differences in definitions, say for 'atheist'.

Why are believers, fundies especially,  missing this crucial component of thought which also, by the way, is essential for critical thinking? There are a number of reasons. For example, one may be that the honing of such skills was never possible, for example if they came from a home schooling environment. Another, more controversial, is that the very neural pathways that predispose them to belief or faith, are also the ones that inhibit predictive ACH conceptualizing. Lastly, it's feasible they may have suffered head injuries, maybe from a DI - using a pugil stick-- in the Marine Corps at Parris Island, and they never got their senses back together.

Sadly, if any of these is true we can be sure they will always take the simpleton path in their thinking as regards atheists and atheism.

---

[1]  George Smith, Atheism - The Case Against God, 1989, Prometheus Books, p. 9.