Looking At Complex Power Series

With complex power series we move one more step toward Laurent series and then calculus of residues.  Terms of a complex series depend no the complex variable z. Most such series are in the form of power series, such as:

å^¥ _{n = 0}   z _n  =  1 + z + z² + z ³ + …..

We understand that in many cases such series will only converge if z is confined to a certain region. In the case of the series above, it converges (according to the ratio test), provided that ÷ z÷ < 1. 

In effect, the above power series converges for all points inside a circle of radius R = 1. This is called the radius of convergence.  This concept of radius of convergence can be applied to every power series. Thus, if a power series is convergent on a circle of some radius r then it is absolutely convergent everywhere inside this circle.  The key point is that R must be finite in order for us to assess a convergence. Else, we say the series diverges.


New Example: 

Consider the complex series:  å^¥ _{n = 1}   (z- a) ⁿ  / n 

Using the ratio test we look at:

lim _{n ® ¥}      [(z- a) ^{n +1}   / n + 1 /   (z- a) ⁿ  / n ]

= (z – a) lim _{n ® ¥}   [n/  n + 1]  =  ÷ z – a ÷

We thereby have convergence for all z such that: 
 ÷ z – a ÷ < 1

And we have divergence for all z such that:  ÷ z – a ÷ >  1

 We have ambiguous or no conclusive test result for: ÷ z – a ÷ = 1

Example (3):    å^¥ _{n = 1}   a ⁿ  (z- a) ⁿ 

Apply the ratio test to write: lim _{n ® ¥}      [a ⁿ⁺¹ (z- a) ^{n +1}    /  a ⁿ  (z- a) ⁿ ]

=  ÷ z – a ÷  lim _{n ® ¥}   [a ⁿ⁺¹ /  a ⁿ ] =   ÷ z – a ÷ × L

We have convergence for all z such that:

 ÷ z – a ÷ × L < 1 or ÷ z – a ÷  < 1/ L

We have divergence for all z such that:

÷ z – a ÷ × L >  1 or ÷ z – a ÷  >  1/ L

No test result for:  ÷ z – a ÷ =   1/ L

Using Taylor’s Theorem in the complex context:

Recall Taylor’s Theorem:

 Let f be a function that is continuous together with its first n + 1 derivatives on an interval containing a and x. Then the value of the function at x is given by:

f(x) = f(a) + f’(a) × (x – a) + f” (a)/ 2! (x – a)² + f”” (a) / 3! (x – a)³ +.......

R_n (x,a)

Where  R_n (x,a)  denotes the remainder.

This can also be applied to the case of complex functions. If f(z) is analytic inside a circle C with center at a (see diagram) then for all z inside C:

f(z) = f(a) + f’(a) × (z – a) + f” (a)/ 2! (z – a)² + f”” (a) / 3! (z – a)³ +......

R_n (z,a)

By reference consider the diagram shown.  Here, we let z be any point inside the circle C. We next construct a circle C1 with center at a, and which encloses z. Then by Cauchy’s integral formula:

f(z) =  1/2 pi  ò_C1  f(w) dw /(w – z)

 One can also show, using appropriate substitutions, that the remainder:

R_n (z,a) =  1/ 2 pi ò_C1  (z- a / w – a)ⁿ f(w) dw /(w – z)

And: lim _{n ® ¥}   R_n = 0. Interested readers are invited to show this by taking:

1 / w – z  = 1/ {w – a) – (z – a) = 1/ (w – a)[ 1/ 1 –(z –a)/(w – a)]

then expanding into terms of a complex series and substituting into Cauchy’s integral formula.

Problems for Math Mavens:

1) Confirm that for the power series: å^¥ _{n = 0}   z _n 

The radius of convergence is R= 1. (I.e. apply the ratio test to show this)

2) Consider the complex power series:

å^¥ _{n = 0}   (z) ⁿ  / n! 

Show that the radius of convergence R = ¥

3) In the Maclaurin series we use the Taylor series but with a = 0. Use the Maclaurin series to expand the function f(z) = exp(-z) to show that the radius of convergence R = ¥.

 

What Would Nelson Mandela Have Done About the NSA?

If Mandela was U.S. President - would he have allowed the NSA to metastasize to what it is now? Hardly!

As the world mourns the death of Nelson Mandela, many tributes will be paid, and achievements recalled. As on Rachel Maddow last night, most of us will finally be brought up to speed on a nation's history we may not have known: that of the South African Apartheid Security state - dedicated to protect the Afrikaner from any contact with those viewed to the next thing to terrorists.

A few of us, mainly in the realm of deep politics, will recall Mandela's signature role as a freedom fighter and how he was persecuted by the S.A. national security state. Working as a lawyer, Mandela was repeatedly arrested for seditious activities and, with the ANC (African National Congress)    leadership, and was initially  unsuccessfully prosecuted in the Treason Trial from 1956 to 1961. Although initially committed to non-violent protest, he co-founded the militant Umkhonto we Sizwe (MK) in 1961 in association with the South African Communist Party, leading a sabotage campaign against the apartheid government. In 1962 he was arrested, convicted of conspiracy to overthrow the government, and sentenced to life imprisonment in the Rivonia Trial.

This is important to learn because in order for us to parse an "alternative history" - say if in some parallel universe Mandela had somehow become President of the U.S. instead of Obama (back in 2008). Then asking: What would he have done with the National Security State with which he'd been saddled - given his background, not as a "constitutional law professor" but an actual freedom fighter?

We know the following already about Mr. Mandela - which many of his latter day idolizers seem to have forgotten:

1. Mandela was a political activist and agitator. He did not shy away from controversy and he did not seek — or obtain — universal approval. Before and after his release from prison, he embraced an unabashedly progressive and provocative platform.

2. Mandela blasted the Iraq War and American imperialism. Mandela called Bush "a president who has no foresight, who cannot think properly," and accused him of "wanting to plunge the world into a holocaust" by going to war in Iraq. "All that [Mr. Bush] wants is Iraqi oil," he said.  He also saw the Iraq War as a greater problem of American imperialism around the world. "If there is a country that has committed unspeakable atrocities in the world, it is the United States of America. They don't care," he said.

3. Mandela called freedom from poverty a "fundamental human right." Mandela considered poverty one of the greatest evils in the world, and spoke out against inequality everywhere. "Massive poverty and obscene inequality are such terrible scourges of our times — times in which the world boasts breathtaking advances in science, technology, industry and wealth accumulation — that they have to rank alongside slavery and apartheid as social evils," he said. He considered ending poverty a basic human duty: "Overcoming poverty is not a gesture of charity. It is an act of justice. It is the protection of a fundamental human right, the right to dignity and a decent life," he said. "While poverty persists, there is no true freedom."

4. Mandela criticized the "War on Terror" and the labeling of individuals as "terrorists"  without due process. On the U.S. terrorist watch list until 2008 himself, Mandela was an outspoken critic of President George W. Bush's war on terror. He warned against rushing to label terrorists without due process. While forcefully calling for Osama bin Laden to be brought to justice, Mandela remarked, "The labeling of Osama bin Laden as the terrorist responsible for those acts before he had been tried and convicted could also be seen as undermining some of the basic tenets of the rule of law."

5. Mandela embraced some of America's biggest political enemies. Mandela incited shock and anger in many American communities for refusing to denounce Cuban dictator Fidel Castro or Libyan Colonel Muammar Gaddafi, who had lent their support to Mandela against South African apartheid. "One of the mistakes the Western world makes is to think that their enemies should be our enemies," he explained to an American TV audience. "We have our own struggle." He added that those leaders "are placing resources at our disposal to win the struggle." He also called the controversial Palestinian Liberation Organization leader Yasser Arafat "a comrade in arms."

6. Mandela was a die-hard supporter of labor unions. Mandela visited the Detroit auto workers union when touring the U.S., immediately claiming kinship with them. "Sisters and brothers, friends and comrades, the man who is speaking is not a stranger here," he said. "The man who is speaking is a member of the UAW. I am your flesh and blood."

(Source: ThinkProgress and the Center for American Progress Action )


Let's assume then, for the sake of argument, the parallel universe would also feature a U.S. of A. that had undergone 9/11, and also had a Bush Jr. who started wars in Afghanistan and Iraq.  However, by some anomaly Nelson Mandela - under different constitutional election laws- became President of the U.S. But this Mandela was Nelson after just being released from Robben Island - so was still in full possession of his faculties at age 72, as opposed to age 90 (in 2008).

The first thing Mandela would certainly have done is pulled all troops out of both Afghanistan and Iraq - especially after just being awarded a Nobel Peace Prize in that parallel cosmos. There is no way, especially after an economic crash such as occurred in 2008, he'd have been able to justify squandering hundreds of billions a year on military adventures while tens of millions lost jobs, children went to bed hungry for lack of enough food, and genuine domestic security unraveled.  The next thing he'd have done is to cut back national security funding that the Bushies had fueled by using "terrorism" as a bogeyman to scare citizens into nodding compliance with surrendering civil liberties.

In particular, he'd have been flabbergasted to find out about the COG (continuity of government) program which under the Bushites  equated political dissent with treason. To him, this would have been contrary to everything he stood for, fought for in his homeland.   He'd have been further infuriated on learning of the "Main Core' program - especially after learning the Fourth Amendment and all others in the Bill of Rights. He'd have pulled back after seeing that:

"without a warrant all the following can be collected: the e-mail addresses you send to and receive from, and the subject lines of those messages; the phone numbers you dial, the numbers that dial in to your line, and the durations of the calls; the Internet sites you visit and the keywords in your Web searches; the destinations of the airline tickets you buy; the amounts and locations of your ATM withdrawals; and the goods and services you purchase on credit cards. All of this information is archived on government supercomputers and, according to sources, also fed into the Main Core database."

The final straw for Mandela would have been learning that up to the time he entered office,  8 million Americans were listed in Main Core as potentially suspect.

And: "In the event of a national emergency, these people could be subject to everything from heightened surveillance and tracking to direct questioning and possibly even detention.”

With recollections of the South African national security state, demanding papers for everyone's travel and assembling massive file data bases on ANC members, families, Nelson would have disassembled COG in a heartbeat. In the next step, he'd have issued an executive order to disband Main Core and the other nefarious additions under the "Patriot Act". It is not beyond possibility he'd have attempted to derail or repeal the Patriot Act itself. At the very least when it was learned how the Bushies used warrantless searches under certain provisions of the Act, he'd have vigorously sought to have them overturned - or at least have vetoed any new "law" passed by a cowed congress, say that reverse-legalized them.

In this parallel universe neither PRISM or XKeyscore would have materialized.  Nor would the mass spying on our allies, which Nelson would have viewed as disreputable and unworthy of a superpower that embraced - or claimed to - principles of liberty.  Finally, the recent revelation about 5 billion cell phone calls a day being compiled would never have been disclosed by an Edward Snowden - indeed it is doubtful there'd have been an Edward Snowden - given Mandela wouldn't have let the agency hire millions of private  sub-contractors.

According to the latest NSA hijinks:
http://www.theguardian.com/world/2013/dec/04/nsa-storing-cell-phone-records-daily-snowden

"The spy agency is said to be tracking the movements of “at least hundreds of millions of devices” in what amounts to a staggeringly powerful surveillance tool. It means the NSA can, through mobile phones, track individuals anywhere they travel – including into private homes – or retrace previously traveled journeys.

The data can also be used to study patterns of behaviour to reveal personal information and relationships between different users."

 Again, being conscious at the most personal level of what personal liberty means Nelson Mandela,  as U.S. President , would never have countenanced such an unimaginable violation of it worldwide, and would have done all in his power to ensure the national security state never attained such power after being elected.

In the parallel universe of December, 2013, Mandela would still have died - but it would have been out of office, and the America for which he served as a steward would have seen greater prosperity by not squandering blood and treasure on unwinnable "wars". That money instead would have been available domestically - to serve the people's needs - as opposed to the warmonger, national security state.

Would the scourge of Neoliberalism still exist in this parallel universe?

Undoubtedly! But given Mandela's background, he'd have wanted no part of it. He knew that no autonomous, "free" markets existed, and that so long as huge corporations dominated and exploited mergers,  markets would tend to be coercive.  As a man of the People, Mandela would know he couldn't also be a Neoliberal or remotely acknowledge "business as the rightful ruler of the country".

Any country!

'Fellowship of Catholic University Students' (FOCUS) Is Out to Lunch on ACA

Would one of the FOCUS women rather be emulating this secular teacher at a Catholic HS, or accepting an unplanned pregnancy because of a rat bastard doing a "roofie-rape"? Inquiring minds want to know!

According to the news in yesterday’s  Denver Post (‘Suit Filed Over Birth Control’ p. 2A), a Catholic college group (FOCUS – or the Fellowship of Catholic University Students)  plans to challenge  the constitutionality of the Affordable Care Act health coverage mandate on the basis of its “forcing religious employers to offer sterilization and contraceptive services, and abortion-inducing drugs for employees.”

 

Mouthpiece Mike Norton of the “Alliance Defending Freedom” (a fulsome name if ever there was one) bloviated yesterday that:

 

“Faith-based organizations should be free to live and operate according to the faith they teach and espouse.”

 

Adding:

“If the government can fine Christian ministries out of existence for keeping their faith there is no limit to what freedom they can take away.”

Of course, this is absolute nonsense. There is no attempt whatsoever to “fine Christian ministries out of existence” – only to have them adhere to the standards of all other employers in the public sphere. If then the Catholic Church operates as an employer in the public sphere – never mind it’s a religious institution – then it’s bound to adhere to public standards pertaining to the Affordable Care Act, which all other public employers at other institutions, corporations are also bound to accept.

Hence, the only thing being “taken away” here is the Church employer’s ability to take away secular employees’ rights to the same standard of health care as all other secular employees’ in the public or private sphere.  The Church, meanwhile, is quite free to morally legislate its own members’ do’s and don’ts to its heart’s content. NO one is taking away that right. 


If the Church sees fit to deny its own members’ as employees access to birth control or the morning after pill, or abortion, then fine. It is well within its purview. But it can’t extrapolate that to secular employees and retain tax-free status as a peculiarly religious institution.  Here's another reason this applies: Unknown to many Catholic purists, many secular hospitals have been taken over by Catholic ones. Their original secular workers who had benefits under those private secular hospitals now see them threatened because of antiquated Church dogmas invoked after the corporate takeover.

According to the Post (ibid.):

“The legal group filed a lawsuit in Denver district court on behalf of the Fellowship of Catholic University Students- a non-profit organization founded in 1998.”

Here's some more of the folderol in their statement(ibid.):

“The (ACA) mandate violates the Religious Freedom Restoration Act, as well as the First and Fifth Amendments to the Constitution.”

Which, again, is total bollocks. There is no such violation because Catholic employers – whether in schools or hospitals or whatever – are quite free to set whatever moral standards they wish for their OWN Church members who are employed in such venues. What they are not free to do is to set those standards for secular employees! Not if they wish to continue to receive  tax-free status from the federal state. As for violating the First and Fifth Amendments, that’s totally preposterous: there is no proscription against freedom of speech – they can spout whatever medieval nonsense that they desire. Nor is there proscription against being a witness against oneself.

 

One of the ignorant First Amendment arguments usually employed is that: "No one is denying the secular women access to contraception or abortion if they want it, they just don't wish to be forced to pay for drugs that violate their religious beliefs".   The point is the "drugs" aren't doing that because they're not being used by the Catholic women who are free to follow their own mores.  They're being used by secular workers as part of the coverage promised under the ACA  health insurance plan!  And, moreover, those drugs - such as contraceptives - may well be needed for more than birth prevention (though that ought to be important enough - given too many mouths to feed can tilt a family over to food stamps, welfare).   But as Sandra Fluke noted in her testimony in March last year: e.g. 

Contraceptives can be used to treat ovarian cysts. If this is not done, the contraceptives denied, then an ovary may need to be surgically removed- and health care costs explode beyond what they would have been.

Contraceptives are also key in limiting severe health problems - such as miscarriage or stroke - which could ensue if a woman becomes pregnant. Moreover, once one commences use of  contraceptives then they need to be taken regularly or they don't work - an educational point that had to be reinforced on dimwit Repukes during their 2012 'war against women'.

 

Meanwhile, we also behold this drivel from John Zimmer – FOCUS VP -  that:

“FOCUS sincere religious beliefs forbid it from providing insurance coverage that includes birth control for its employees, it says: ‘ As Catholics we do not impose our faith on anyone but propose to others the life-changing power of God’s love.”

 

But as I noted in my recent book (‘Beyond Atheism, Beyond God’), humans with their limited neural capacity aren’t in any remote position to define or know “God” since the noun (G-o-d) isn’t the same as the entity. Since the entity is supposedly “infinite” then no finite brain can comprehend it. The best it can do is forge approximate god-concepts which are relative, limited and subjective.  Hence the term “God’s love” is totally meaningless other than on a subjective level. It’s vacuous verbiage because if one can’t comprehend “God” one surely can’t comprehend any “love” derived therefrom.  And by extension one is proposing to others nothing more than one’s own ideations as internally subjective lingo.  As Philosopher Joseph Campbell put it (The Power of Myth, p. 56):

 

“ 'God’ is an ambiguous word in our language because it appears to refer to something that is known..”

 

All of which reinforces the point that when people use the word G-o-d they’re not talking or writing about an actual entity but a limited construct or ideation configured as a noun, which we call a God concept. People BELIEVE they are referencing the real entity by use of the noun but they are deluding themselves. It would be analogous to me going to a restaurant – seeing a ‘steak’ picture on the menu – then tearing the image out, consuming it and asserting I’d eaten a ‘steak’.  Any such pretense would ordinarily earn one a fast trip to the loony bin.

 

As for the claim of “not imposing their faith on anyone”, I am afraid FOCUS’ collegians need to take a semantics or logic course, preferably both.  If indeed, they’re all about preventing SECULAR employees – say atheists like me – from accessing artificial  birth control- then they are indeed imposing their faith.  Besides - what if the tables were turned? How would Catholic purists and dogmatists react if a Hindu-run corporation took over a Catholic Hospital? Would such a happenstance be called a "forced conversion" or  "violation of religious liberty"? OR - would it be considered a business transaction only? Also, if the newly merged entity produced subsidiary corporations - what religious principles would rightly apply? Catholic blowhards and tools like FOCUS need to think such questions through before they get all sanctimonious on us.

 

Now, if instead they confined their natural law moralism to Catholic employees only, there’d be no objection. “Imposing one’s faith” means taking away rights of secular citizens– based on invoking one’s own imagined religious principles.  In other words, extrapolating their “principles” beyond their proper moral domain to take away the rights of citizens who aren’t part of that domain other than in an employee capacity.

 

While we’re at it, let’s take a closer look at these principles which they profess to hold so dear. Biologist Elizabeth A. Daugherty  has asked ('The Lessons of Zoology'. in Contraception and Holiness, p. 110):

"Why do we call secondary the ends of the sexual act which have been accorded in fullness to us, and why do we call primary the end which we share with the lower animals?"

She's referring to the fact that the core of Pius XI's  original encyclical Casti Connubii was that the "sin" of artificial contraception inhered in making primary a sexual aspect that in reality is only "secondary". According to that pontiff:

"Since therefore the conjugal act is destined primarily by nature for the begetting of children, those who in exercising it deliberately frustrate its natural power and purpose sin against nature."

Which is irredeemable codswallop. As Daugherty notes in her chapter (op. cit.)  what the pontiff and his ilk really sought to do is reduce humans to the state of lower animals, at the behest of their "natural" reproductive cycles. In this sense, unlike the lower animals, humans have the intellectual capacity and sense of novelty to introduce a vast variety of pleasure-play into their sex relations. They aren't yoked to  primitive instincts to simply mount and hump at specific times. As Daugherty notes (pp. 96- 97):

"After ovulation, all mammalian females are under the influence of progesterone from the corpus luteum. This is a period of rapidly declining estrogenic activity which ends the sexual receptivity of the lower mammalian female, whether or not fertilization occurs.
 

But (in humans) marital relations continue during this progesterone -dominated period before the abrupt onset of menstruation. It is the period of lowest estrogenic activity and the progesterone-dominated period after ovulation which are known as the 'safe period' for marital relations."

This then, is what the Catholics'  “rhythm method” (of  Ogino-Kaus)  seeks to do: establish the "safe period" for a particular woman and then ordain that this is the time to safely have sexual relations if one wishes to not have any kids. The trouble is, it requires meticulous temperature taking at various times during a cycle to establish where that safe period begins and ends, and often this will be for no more than 10 days or so in a given month. Presumably, the couple is quite happy to do without sex the other two thirds of the time!

Thus, the moralizers of the Vatican are actually demanding that married couples act UNNATURALLY, since as Daugherty observes (ibid.):

"Humans are free from physiologically determined sexual desires so we possess a more or less permanent sexuality from adolescence to old age."

Indeed! But the Church and its robed minions (and now evidently younger acolytes in colleges)  seek to dictate that despite being sexual or having sexual desires from adolescence until old age, her members are only free to discharge those desires under certain limited times and limited conditions. For example, teens who have such exploding desires on account of their hormones are warned they cannot even masturbate to relieve themselves because those organs are only allowed to morally function in the state of marriage. Then....once married, the couple is informed they may gratify their mutual sexual desires only if they are open to conception....that is, unless they use the rhythm method.  Anyone not see a pattern here?

 

But once more, my gripe is not so much the Vatican roping  in its own flock’s gonads, but seeking to rope in the gonads of secular employees working at their schools, hospitals etc. (And please don’t anyone tell me “They can get another job!” – NO, not in this low aggregate demand, high unemployment environment. That’s like telling a guy who’s been denied food stamps (and on his last legs)  to go jump off a roof, then he won’t need them!) Besides, would Catholics tell Catholic employees to just "get another job" if their hospital was taken over by Hindu owners? I doubt it!
 

Julian Pleasants has observed (op. cit., p. 88) the Vatican has always been hostage to:

 

"Aristotelian modes of thought which tend to fix behaviors within very limited and fixed definitions and categories."

Thus, the Church once believed it "natural" that some men be enslaved because they were “unable to manage their own affairs”  (ibid.)So why be surprised when the same Church seeks to ordain all her members abide by a sexuality more fitting of lower primates?   However, be that as it may, we MUST express surprise (and outrage ) when the Church and its enablers like FOCUS seek to impose that same sexuality on secularists. That is simply unacceptable.

 

The abiding question for me is: Why would 21^st century college students, embedded in a bunch like FOCUS,  be yoked  to Aristotelian modes of thought they seek to impose on their more enlightened peers or secularists? Maybe they are jealous they themselves never broke free of their mental chains to see just how limited they are by the Vatican’s nonsense. And while we’re at it, let me remind them that the birth control proscription has never been stated ex Cathedra so it’s not infallible. It is proposed under the Magisterium or Teaching Office. As one honest padre once confided to me: "You are free to use your own conscience in this regard, since the teaching isn't binding."

 

As for abortion, the FOCUS groupies ought to bear in mind that John Connery, S.J,  a leading Catholic historian,  showed that up to 1869 the Church’s Canon Law had historically held abortion to be murder only subsequent to the end of the first trimester.[1]  The interesting thing is that for the bulk of Church history the practice of abortion was allowed, at least up to the first three months of pregnancy. More interesting is that the doctrine of papal infallibility was pronounced in 1870. Could it be that the latter doctrine was introduced to back up the Church’s change in its moral position on abortion? If so, one wonders who really controls the Church, the Holy Spirit-  as is so often claimed-  or zealous prelates in positions of high power who fear relinquishing vise-like grips on their congregations[2]?

Oh, one also wonders if they still teach critical thinking at Catholic universities, as Loyola did in the 1960s - even encouraging atheists such as Jean -Paul Sarte to lecture us on existentialism! To see FOCUS' antics is to wonder if critical thinking went the way of the Dodo - at least in the universities that harbor FOCUS outlets!

---

[1] Sagan and Druyan: PARADE, (April 22, 1990), 6.

[2] And  (most recently), nuns, mainly belonging to The Leadership Conference of Women Religious, which was issued a reprimand by the Vatican for straying outside the purview of Church teachings – especially after they defended the inclusion of contraceptive services in Obama’s Affordable Care Act. The Vatican has also censured Sr. Margaret Farley (of Yale Divinity School) after publication of her book, ‘Just Love’. Farley evidently stepped on Vatican toes when she justified artificial contraception as well as masturbation within marriage as an aid to preserving fidelity.

 

Could Comet ISON's Demise Lead to Solar Physics Insights?

ISON  (left) taken on Nov. 25 by NASA's Stereo-A spacecraft as it approaches the Sun.

In the aftermath of the disruption of Comet ISON, the question now becomes whether we can salvage scientific information, especially on the Sun's corona.   Undoubtedly, any break up detected and 'unpacked' over its sling shot bypass will also divulge information on the comet composition itself - but my interest is in the value in terms of solar revelations.

The fact is that the anatomy of coronal solar loops is a key subject of interest, including to solar flare forecasting.   To fix ideas some of the physical-structural details in a typical solar loop are shown below:


We can observe the dynamics of such loops using a variety of space-based instruments, but they are still not fully understood. For example, the central issue of which magnetic field lines loop back on the Sun (as depicted) and can contain the million plus degree plasma, and which are forced open in the heliosphere (the region above the solar surface through which the solar wind extends) is still not resolved. While we can analyze the magnetic topology and field line behavior to an extent we can't explicitly tie them to particular solar active regions. One major reason is because they occur in a domain (the lower corona) from which light is hardly emitted and spacecraft can't venture (they'd be incinerated).

As shown in Fig. 2,  the typical solar loop is anchored in a more dense medium in which convection occurs. To see convection in action, get a transparent (glass) saucepan and fill it halfway with water, then heat it to boiling. Observe laterally through the glass when boiling point is attained. Elements (bubbles)  of water are heated near the flame, then rise on expansion to the top, are cooled, then descend back to the source of heat. This occurs in analogous fashion in the Sun except we are talking about elements of solar plasma.  The overturning plasma and small-scale field bundles (entrained within it) supply non-radiative or 'convective' energy to the solar atmosphere. The dissipation of that energy then heats the outer atmosphere (corona) to 1-3 million degrees Kelvin or some 300 times hotter than the temperature of the solar surface (photosphere) itself.  This outer atmosphere then radiates in the high frequency band of the electro-magnetic spectrum, namely in x-rays and EUV (extreme ultra-violet).

Fortunately, we aren't completely hamstrung because nature has provided us with natural probes: sun-grazing comets.  For example, in July, 2011, the comet designated C /2011 N3 or 'N3' for short, transited the solar corona and its tail lit up in the extreme ultraviolet. N3 passed to within 110,000 km of the solar surface, losing up to 100 tons per second of its mass as it did so. A half year later, comet C/2011 W3 ('Lovejoy') passed to within 135,000 km of the surface.  Most noteworthy, the comet's tail was observed moving in response to forces exerted by both the Sun's magnetic field and its atmosphere.

What is the critical key to a useful sun-grazing comet? Primarily it's the mass. While the deep corona is way too harsh an environment for spacecraft, the large initial masses  of sun-grazer comets often enable them to survive long enough to deduce something about the coronal environment.  In this context, we look at the so-called Lorentz force: F = q(E + v X B)  which acts on ionized cometary material (i.e. with gases that have lost one or more electrons).  Then, the resulting ion motions reveal the local orientation of the coronal magnetic field (which shapes the loops).  This is possible even as the comet's ions decelerate and settle into the coronal plasma.

What physical factors can we use to predict the comet tail's evolution, say as a useful indicator of coronal field lines? One is the ratio of the energy density of the coronal magnetic field,( B² / 2u_o ) to the kinetic energy density of the plasma in the comet's tail (i.e.  0.5 r v²).   For example, in the case of Comet N3, the comet's inertia dominated, so as the cometary plasma decelerated during collisions with the corona, the latter's magnetic field became strongly deformed.  By contrast, in Lovejoy's case, the solar magnetic field - reflected in its energy density - appeared to hold its own. Hence, each comet delivered key information on the behavior of the corona in relation to ambient solar factors.

In addition, information was gleaned from the varying deflections of Lovejoy's tail- showing a highly inhomogeneous medium. Subsequent state of the art computer modeling of the solar corona confirmed the interpretation and revealed a striking consistency between observed tail and the orientation of the magnetic field in the solar corona. In other words, Lovejoy really came through as an excellent solar probe -  and provided critical knowledge for projecting future behavior of solar loops.

But let's not get ahead of ourselves. The ultimate goal is to create or construct a global model of the corona. But for this we need a full, 3D spherical map of the Sun. At the present time (or at least until we can get a solar orbiting space craft) only the field in front of the Sun can be reliably measured and then only to latitudes between 70 degrees north and south heliographic. The Sun's rate of rotation is roughly 27 days meaning that for up to two weeks our observational access is blocked until the given active region comes back into view.

Given, this only about one fourth of the solar surface can be accurately mapped from observations of its magnetic field. In effect, no matter how many 'ideal' comet probes we can make use of, we can't get a genuine handle on the corona until we get a full sphere map of it. Maybe when the "war" in Afghanistan is finally ended, and we bring those troops back home, we can use some of the money saved to launch a solar orbiter.

Despite Lovejoy's short (2 -3 day) survival after passing through the solar corona, it did behave enough as a probe to permit a computer model to confirm certain properties of the corona.  What about Comet ISON?  Unfortunately, in terms of learning more about the corona itself, ISON is of limited value. ISON's perihelion distance of 1.2 million km put it much farther out in the corona than Lovejoy's (135,000 km) passage.  Since plasma density drops rapidly with altitude in the corona  ISON's solar interaction will be more dominated by waves and turbulence. 

In this case, it might be possible for ISON's demise to provide us with some useful information about ion acoustic and Alfven wave behavior (see e.g. http://brane-space.blogspot.com/2011/04/getting-handle-on-alfven-waves-1.html

 at the level of the high corona. To check this, we could make use of modeled solar parameters at that altitude, say the coronal pressure, P = 2 n T (where n is the number density and T the temperature), as well as the thermal conductivity: k  = k _o  T ^5/2   and  thence obtain the coronal heat flux density           q = -k grad T and work these into appropriate wave equations, to check for consistency with computer models.

To what extent this can be done from ISON data remains to be seen, but I am confident we can at least extract something of value to further our understanding of the Sun's corona - at least at the level that ISON transited.

U.S. Adults & Students Bomb Out In Math, Science ....Again!

Conducting a workshop in August, 1978 to help Caribbean teachers with the CXC Astronomy syllabus

One of the amusing comments of Caribbean teachers attending a CXC (Caribbean Examinations Council)  astronomy workshop in August, 1978 in Castries, St. Lucia was:

"Why is there so much physics and math in this syllabus content? Are we expected to be able to integrate both these subjects?"

Well, uh, yes .....that's why this subject is called Integrated Science and why astronomy is taught for one trimester as part of it. After doing some basic background review, however, most of the teachers grew a bit more comfortable.  Though astronomy ultimately became defunct after five years - because fewer and fewer teachers were prepared to handle it - the seeds sown in the inaugural year allowed a vast number of science teachers to enhance their physics and math skills.

Could a deficit in physics and math background, skill also explain why U.S. students continue to not do so well compared to other advanced nations? According to recent results of a world wide test - administered to over 156,000 people in 20 nations, between the ages of 16 and 65 - the conclusions are not sanguine, for the students or the adults. In the case of the high school students they placed 31st in math, and 24th in science, not having changed position in the last decade.

Meanwhile, the students of Shanghai, China (the nation that's just dispatched a space craft to the Moon) scored highest, with their results equivalent to at least one additional year of school - over U.S. competitors. Nor was this simply a matter of regurgitating material, but instead the students demonstrated they "were very good at higher order skills reflecting what you can do with what you already know."

One example of this type of problem would be:

"A group of 4 astronauts lands on Mars with solar radiation collection material of total area 2000 m^2. If the efficiency of the material is 30%, and the ambient night time temperature on Mars for their base location (Isidis Planitia) is -40 C (10 C day time high), will they have adequate collecting material if the solar constant on Mars is 620 W/m^2? (Assume insulating material with a thermal conductivity of 0.08 W/mC, and a need to keep the inside area of their domecile at least at 10 C, requiring solar radiant energy collected of at least 1,200 W per minute for an area of 10 m x 10 m.)

Estimate the thickness of insulating material they're likely to need in order to make it work. Comment on whether this expedition is even feasible given the limits of their materials, and that no more than 100 m^3 of insulating material can be taken."

That Chinese students are proficient at doing this type of critical thinking application problem and U.S. students mainly are not, ought to be severe cause for worry. In addition, this divergence can also explain why so many in the U.S. still fail to grasp the basics of climate change, including why spates of frigid weather and snow do not contradict it.  This take is also reinforced by the fact that American adults scored below the international average on a global test in math, reading and problem-solving using technology — all skills considered critical for global competitiveness and economic strength.

Meanwhile, adults in Japan, Canada, Australia, Finland and multiple other countries scored significantly higher than the United States in all three areas on the test, according to results released Tuesday by the Organization for Economic Cooperation and Development, which is made up of mostly industrialized member countries.  Are we a nation in profound decline? I submit to you that given the results on multiple levels, we are, and there's no gainsaying it.

How complex were the problems? Maybe you are thinking they were similar to the sample one shown above. Not so! Beyond basic reading and math, respondents were tested on activities such as calculating mileage reimbursement owed to a salesman, sorting email and comparing food expiration dates on grocery store tags.  None of which require profound insight or genius level skills, or shouldn't. But recall that a similar test given in the 1980s disclosed adults who: didn't know the Sun was a star, and believed the Sun moved round the Earth, not vice versa.

Not only did Americans score poorly compared to many international competitors, the findings highlighted the gap between American high- and low-skilled workers, and how hard it is to move ahead when your parents have not. To fix ideas,  in both reading and math, for example, those with college-educated parents did better than those whose parents did not complete high school.

Most worrisome, the U.S. tested below average on literacy. Coupled with the nation's "large social disparities, the test's findings reveal that for Americans, social background has a major impact on literary skills," the study said.

As we confront an era with ever more advanced technological issues, not only climate change but (very soon) the advent of quantum computing and bio-engineering, one wonders just how prepared Americans are. While too many still remain mesmerized by the bubbling DOW, they'd do much better to ensure they attain essential critical thinking skills - for themselves and progeny.

A set of sample math and science questions to see how you fare:

1)  In a given week a salesman travels 134 miles, during which he pays an average of $3.29 per gallon for gas. If his Buick Enclave gets 12 miles to the gallon, how much should he be reimbursed?

2) You're given a straight rule, a doughnut and a circular disc. Draw a sketch of each and mark the center of gravity in each sketch. What is meant by the center of gravity?

3) A girl is given 100 cubic centimeters of boiling water in a beaker, to which 50 grams of  ice is added. Using a thermometer, how will she know when the mixture has reached thermal equilibrium?

4) One Big Mac and Cheese with condiments has 3,800 calories. You want to burn them off with exercise, and choose jogging. This allows 150 calories to be burned off per half hour,. How long will you need to jog to burn off the Big Mac?

5) An air conditioner consists of a fan coil unit and a compressor unit.  When switched on, the fan coil unit rated at 250 watts, operates continuously. The compressor unit - rated at 1.6 kilowatt- operates only 40% of the time. If the air conditioner is switched on for 12 hours a day, find the total electrical energy in kwh (kilowatt-hours) consumed in a 30-day month.

6) Find the value of x, if:

6x - 24 =   12 - 3x


7) If seven  pencils and 5 erasers cost $11.60, but 5 pencils and 3 erasers cost $7.60. Find the cost of 8 erasers.

8) The floor of a room is in the shape of a rectangle. The floor is c meters long. The width of the floor is 2 meters less than its length. State, in terms of the unknown c:

i) the width of the floor

ii) the area of the floor

If the area of the floor turns out to be 15 m²    then find the width of the floor in meters.

Time to End the Afghan Intervention - And Use the $$$ HERE in the U.S. of A.!

U.S. Marines urinate on Taliban corpses almost 2 years ago. It is now time, with the opportunity to leave -  to do so. We can't afford to pour $8 billion a year into this deficit pit to stay another 10 yrs.













With the news of an 8 year old boy killed Friday in Kandahar province, adding another to the list of thousands killed by wayward drone attacks in Afghanistan, it is now time to pull ALL U.S. troops out forthwith and the use the money we'd spend there on needs here. I am talking about nearly 16 million American kids who go to bed hungry each night, after the SNAP program was discontinued and now new food stamp cuts are being proposed as part of a "Farm" bill. A Farm bill, btw, which is actually a subsidy to rich agri-business operations, not to little "moms and pops" as some might believe.

Writing about the bill's  $14  billion a year "crop insurance" subsidy, Jim Hightower wrote in one of his recent columns (e.g. Colorado Springs Independent, Nov. 30):

While the House wants to force food stamp recipients to prove that their incomes are low enough to qualify for benefits, the crop insurance handout requires no means testing and has no limits on how much recipients can get. This means that billionaires who're only incidental "farmers" will be among the biggest beneficiaries


We need to process that, as well as the image of hungry kids by the millions who have to go to school each day with empty bellies, as their parents now struggle to get adequate food supplies after the Reeps cut the SNAP program. With the $36 a month cut to SNAP - enough for nearly a week's dinner meals for a family of 4- people now are pouring into food banks which themselves are finding their shelves bare.

Now, how can THIS be reconciled with the willingness to piss 8 billion a year for the next TEN years into Afghanistan, according to a new "accord" the U.S. wants - to allow up to 10,000 American troops to remain there until 2024? The total intended to be pissed away amounts to $80 fucking BILLION - enough to ensure NO American kid or his parents goes hungry for the next ten years. You want to know where the money is to feed those families before you cut the food stamp program by $40 billion, THERE it is, Mr. GOOPr!

Obama originally promised a pullout next year following 13 years of the longest "war" ever, but now for some odd reason is proffering a "security compact" to stay another ten years with billions spent each year. I am wondering if the stresses of his healthcare.gov debacle have overly muddled his neurons or what?  Whatever, Susan Rice insists the Americans will be forced to withdraw by 2015 without a security deal.

Meanwhile, Hamid Karzai has insisted on his own assurances as part of the deal, including no reckless drone attacks on friendlies and no kicking their doors in, invading their homes and doing spurious searches.  The U.S. in response has made it clear it can't make such guarantees, and Obama evidently wrote a letter to Karzai assuring him that "the U.S. will make every effort to respect the sanctity and dignity of Afghans in their homes."

Well, try running that one past the families already  affected, and nearly 15,000 homes kicked in, and occupants tossed face down on the floors,  over 12 years..

Additionally, the U.S. press is all het up about Karzai's "disparity of reaction" (New York Times, Dec. 1, p. 16) reacting with fury to a U.S. led drone attack, but remaining silent about a Taliban suicide bomber attack. But again, this is another reason to leave the damned place and make no deals with Karzai. Indeed, Atiqullah Baryalai, quoted in the Times piece, observed that Karzai's silence is: "sending a message to the Taliban that he really doesn't want a security agreement with the Americans".

If this is the case, we shouldn't walk out, we ought to race the fuck out and leave Karzai and his nation of fifty tribes and tribal passions to themselves. Even remaining 50 years wouldn't be sufficient to help them!

Essentially, Karzai by his dithering on any security deal has given us a perfect excuse to leave Afghanistan and not look back. But what has the U.S. been thinking of doing? Going instead to Karzai's second in command and seeking a deal with him! WHY? You have the opportunity to leave, TAKE IT!

Our people are the ones that need that money! Our kids are the ones going to bed hungry each night, and 48 million Americans are living off piddling, grudgingly bestowed food stamps that can barely keep body and soul together. 

If Obama has any sense he will take the cue on offer, thank Karzai for his input and LEAVE!  Invest right here at home now, for once, instead of pissing $80 b more into a barren deficit trap!

One more thing: Dianne Feinstein has initiated a new round of terror mongering, asserting we are at increased risk now for terror attacks and more than we have ever been. Obviously and clearly she wants more funding for the national security state which has served her and hers just fine. She likely also wants to keep the NSA spooks fully occupied and funded as long as she can bleed money from the rest of us. Of one thing she is correct: that terror risks will rise the longer we remain with our asses in the Middle East, antagonizing all and sundry.

We seem not to have processed that is what led initially to 9/11 as blowback, as the late Chalmers Johnson has noted.  When asked in an interview with In These Times if September 11 was an example of blowback, Johnson responded:

Of course it is. That's exactly what my book was written for: It was a warning to my fellow Americans, a year ago, that our foreign policy was going to produce something like this. It's important to stress, contrary to what people in Washington and the media are saying, that this was not an attack on the United States: This was an attack on American foreign policy.


Similarly, the longer we remain in Afghanistan the greater the risk of blowback. As with Vietnam we need to let them, the Afghans, determine their own destiny.

We can't be the 'Cops of the World', nor can we be nation builders when our own country daily falls into disrepair as the human fabric is also unraveled by misplaced priorities! We need to attend to our immediate domestic security before any more foreign "war" adventures!

No! No Cell Phones on Planes!

Back in July, 2009, en route to my dad's funeral, traveling from Colorado Springs to Atlanta at 7 a.m., I was in no mood for babble after boarding. I just wanted to be immersed in my own thoughts and the occasional music piped into my earphones. I was fortunate to be seated next to two Delta air hostesses (both in their 30s) en route back to their hub in Atlanta. They were perfect seat mates for this sort of sympathy travel: they spoke softly to each other, watched the in-flight movie and let me to my own contemplation until about a half hour prior to landing.  Then the young woman seated next to me asked if I was going on holiday. I replied, 'No, headed for my dad's funeral in Port Charlotte, Florida'. This then spawned a longer conversation about his life, after she expressed condolences. Also, we briefly got into my own interests, that I'd just released a new book etc.

Basic point? No air captive wants to be hear babble unless it's: a) welcome, b) at the right time, and c)  is appropriate to the context. Aside from screaming bratskies on planes, one of the worst experiences the contemporary air traveler can have is to be seated adjacent to a narcissistic endless yapper who doesn't know when to call it quits. Fortunately, my encounters with such bozos are infrequent, given my wife and myself most often travel together.

Now, alas, with the FCC prepared to let airlines permit cell phone use during travel, we may be looking at the advent at the endless babbler  contaminating limited air space with noise pollution. Never mind s/he may not be talking to you. Merrill Markoe ('Shut Up and Fly!', TIME, December 9, p. 23) points out the problem eloquently:

"If seatmates start to babble to you, revealing themselves to be primary narcissists so enchanted by the sound of their own chatter that they'll keep talking even when they're out of topics - a nicely executed icy glare will  often ice them into silence....

Once cell phone calls are permitted, the FCC will be offering these people a behavioral pass. If their fellow passengers shut them down they can turn to their cell phones instead. The people on the receiving end of their calls may not even be paying attention. They might be playing video games and going 'Mmm...Mmm.....hmmm'

Meanwhile, the poor beleaguered seatmate remains stuck, listening to these grating narratives being repeated.."

This is spot-on and calls into scrutiny an observation I've made concerning cell phones, which is to say that once a person gets one he believes he's been catapulted into automatic big shot status. He may not be a Fortune 500 CEO but hey, the cell confers power and when he pulls it out he can assume the same power vibe as the CEO using his to demand his cubed vassals take their marching orders.  Thus, the mere ownership of one almost inevitably incites its use and generally this becomes overuse for any stupid reason whatsoever. In this sense, comedian Louis CK was absolutely right when he provoked angst by noting it's merely one more device to keep people thinking they're "doing something"  - when they aren't doing diddly or squat. (Obviously, emergency situations are different.)

Markoe interjects this on noting that the typical response will be "People need their cell phones now!" They "can't expect to exist hours without them!"   Which is balderdash and plays into Louis CK's point that people without a life (a real life) merely employ them to try to escape their own sense of isolation, loneliness. In the olden days, back in the 60s, people read books when on airplanes - but evidently this is no longer "cool". Now,  one must be seen instead holding a stupid little device in his hand and blabbing away to some poor forlorn fool who probably has an actual life and things to do.

Am I a curmudgeon? Well, ok, I do have curmudgeonly tendencies since having lived nearly seven decades I've seen way too much bullshit, and PR crappola that is exploited when people wish to avoid hard truths. It's easier for too many to live in unreality than reality. But consider this: unless you are in Business Class, flying has become increasingly god-awful. There's less and less room between the seats (typical seat widths are now 17-17.5" compared to 22" in the 60s),  and the pitch of seats is now so restricted you can't  lean backward without asking permission of the guy behind you. Meanwhile, the food is like warmed over kibbles, seat interspaces are like small garbage dumps, and any real food (at least closer approximations) costs extra.  See also: http://brane-space.blogspot.com/2013/07/yes-flying-was-really-much-better-in-60s.html

Now, given this, why in hell would any sane person wish to add over-caffeinated road warriors - jabbering away like pseudo big shots - to this already foul environment?

At least the FCC chairman, Tom Wheeler (and his 4 commission cohorts) only issue rule change proposals, they won't be enforcing any. Thus, each airline will get to decide whether it wants to allow passengers to make cell phone calls. Already Delta, after taking an online passenger poll, has found 70% absolutely do not want it. In a statement issued 2 weeks ago Delta said: 'The overwhelming sentiment among our customers is that cell phones can only make their flying experience worse'.

Exactly! But make no mistake other airlines - less classy and less attendant to passengers' wishes - will seek to exploit this as an opportunity to boost profits. In this case, phone companies would charge you extra for the calls made in flight and airlines would get a portion of those fees.  And make no mistake that airlines will want a piece of the action, because - if they accept cell phone calls- they'll have to install telecom equipment to make calls possible. And that won't be cheap.

Other suggestions have been made that are reasonable and ought to strike a balance. One of these is that airlines ought to ban use of cells for any flights 5 hours or less. This is short enough so that anyone who needs to communicate should be able to do so through email or text message just by connecting to the in-flight Wi-Fi service.

On longer flights, airlines could install sound proof phone booths, perhaps about half the size of the airline rest rooms. Such a move would satisfy both the hyper-connected, wannabe big shot traveler- who thinks he's too important to ever be out of reach - and the rest of us who just want to reach our destinations without going ape shit in the cabin and being arrested as "domestic terrorists" for strangling a cell blabber in his seat.

Let's hope other airlines besides Delta see the light, and look to enhancing their passengers' flight experience (or at least not making it worse). But I wouldn't make book on it!