Thursday, April 30, 2020

Analyzing Parametric Curves In 3-Space

Image result for Helix

The helix - a curve in 3-space to be analyzed.

Curves in 3-space play a major role in many areas of physics and are worthwhile exploring.  Adopting a positive direction for measuring distance along such a curve, say from a point P 0 at t =0, then the arc length can then be measured as P moves from  P 0.   Then the  position of P becomes a function if the arc length  s  from P 0  to P.  Then the vector:

R =  ix  + y + z

is also a function of s.   Of particular interest will be the derivative:

dR/ ds =  i(dx/ds) + j(dy/ds) + k(dz/ds)


By taking the limit:

dR/ ds =    lim  s   -> 0   (Rs)

It can be shown one obtains a unit vector tangent to the curve at P, say, and pointing in the direction in which arc length s increases along the curve. One thereby arrives at the vector T:

T = dR/ ds

Consider for reference the helix shown which is described by:

x = a cos wt,   y =  a sin wt,  z = bt

If we take T = dR/ ds =  i(dx/ds) + j(dy/ds) + k(dz/ds)

We get:

i( - w sin wt  dt/ds)   +   j(cos wt  dt/ds) + k( b dt/ds)

Which is the unit vector to the helix at any point P.  It is also of interest here to determine the magnitude of the scalar dt/ds.  This is accomplished by using the fact that:

| T |  =  1   or  T·T   =   1

Factoring out (dt/ds) from the earlier eqn.  and by using some algebra:

( a 2 2  +  2)  (dt/ds) 2  =   1

Whence we take the square roots, and make the scalar factor the subject, so:

dt/ ds  =   +   1/   Ö ( a 2 2  +  2)    

Since we've agreed s (arc length) is an increasing function of t we can take dt/ds as a positive constant and  write for the tangent vector:

T  =    (- i sin   wt  +  j  cos  wt )  + b k /  Ö ( a 2 2  +  2)    


The length (or arc length) of a given space curve can be found by computing ds from:


ds   =   + Ö (dx2 + dy 2  + dz 2 )  

And integrating between appropriate limits. In this sense the procedure is much like that for obtaining the arc length of a curve in 2 dimensions.   In addition, the curvature  k  of a space curve (i.e. in 3 dimensions) is defined by the same vector equation as for a plane curve.  This means taking the derivative dT/ ds which will be either 0 (straight line or  k = 0) or  will be normal to T.  Defining a unit vector N, i.e. a principal normal to the curve at P, we may write:

| N |  =  1     and:  dT/ ds =       k N

Then the curvature for the helix is obtained:

    |  dT/ ds |    =  [(dT/ dt)/  ds/ dt]  =   


 (- i sin   wt  +  j  cos  wt )  + b k /  Ö ( a 2 2  +  2)  {Ö ( a 2 2  +  2) }

=  (- i sin   wt  +  j  cos  wt )  + b k /  ( a 2 2  +  2)  

Factor out the non-trig term:

dT/ ds =    /  ( a 2 2  +  2)    [  i sin   wt  +  j  cos  wt   + b k ]

Then by inspection (i.e. comparing with  dT/ ds =       k N):

 k  =  /  ( a 2 2  +  2


Example problem:  

Given a curve defined by:

x =  6 sin 2 t,  y = 6 cos 2 t, z = 5t


Find an expression for  | T |   and T in terms of dt/ ds

Solution:

dx/dt=  12 cost 2t, dy/dt = -12 sin 2t,  dz/dt = 5

Therefore:

| T |  =  1 = (dt/ds)2 [(12 cost 2t)2 + (-12 sin 2t)2 + 52]

| T |  =  1 = (dt/ds)2 [(144 + 25)] =   (dt/ds)2  (169)

So that: (dt/ds)   = 1/ 13  

And: 

T   = 1/ 13   [12 cost 2t) i + (-12 sin 2t) j + 5 k ] dt/ds

Problems for the Math whiz:

1) For the example problem:

Find the curvature   and the length of the curve from t = 0 to t =  p

2)  For a space curve defined by:

. x = exp(t) cost t,   y = exp(t) sin t,  z =  exp (t)


Find  the unit vector T =  dR/ ds


Tuesday, April 28, 2020

Examining The Basis Of Scientific Materialism

















Scientific Materialism  - the  philosophical basis of all our modern physics, chemistry and biology- is generally treated as if it's one monolithic entity rather than a diverse set of approaches and views. In this respect, and because of oversimplification, it becomes a more  convenient strawman for religionists of all persuasions to attack.

One reason Materialism gives so much grief to intellectual poseurs is that it is categorizable in so many different ways. Because of the complexity, most simple-minded mentalities just ditch any nuance and opt to portray the cartoon version. The problem is that this choice renders most of what they write irrelevant and insipid, especially when they claim generalities, for example asserting "Materialism commits the identity fallacy by equating mind and brain". Indeed, such general statements make them look like blinking idiots.

In this blog  post I want to clear up some of those misconceptions and also show why one form (Modern or Scientific Materialism) remains the best to economically explain such things as consciousness, and the operation of the brain.

Differentiating Types of Materialism:

Perhaps the earliest form of Materialism was Mechanical Materialism or the theory that the world (and universe) consisted entirely of hard, massy material objects, which, when imperceptibly small, were described as "atoms". This set the stage for the "atomistic materialism" promoted by the Greeks, Leucippus and Demokritos, who used it as a basis for the philosophy of Epicureanism. It may rightly be said, therefore, that this was the precursor of the reductionist/materialist/realist school of thought. This version denies that immaterial or apparently immaterial things (such as minds) exist or explains them away as being material things or motions of material things.

It is very ironic (but true) that a Greek Materialist like Demokritos would have serious problems with modern scientific Materialism and its claims for immaterial (but physical!) fields, atoms and waves! Indeed, more than one observer has indicated these early Materialists would resoundingly deny Modern Materialism!

Wherefore this "Modern Materialism" so unlike its predecessor? The need for it became manifest by the time Ernest Rutherford began his simple atomic experiments, and then Max Planck discovered the quantum (E= hf) ...leading to Erwin Schrodinger's and Werner Heisenberg's depiction of atoms as vibrating wave states of different energy. See, e.g.


This  wave emphasis meant that not only did immaterial entities exist, but they could also be quantified and have physical existence.

Thereby, modern physics conceived matter as made up of such things as electrons, protons, and mesons, which are very unlike the hard, massy, stonelike particles conjectured in primitive mechanical Materialism. Further distinction between matter and energy was also broken down via Albert Einstein's theory of special relativity which showed that energy has inertia. His most famous equation: E = mc 2, became the basis of atomic energy via nuclear reactions - both via fission and fusion.

The explosion of the first atomic bomb at Alamagordo, NM, showed in no uncertain terms this new physics wasn't some airy fairy hogwash but as real as a ten million degree fireball that could roast a human body in two nanoseconds, reducing it to a few drops of goo and fried bones. Thus, the connection from the modern physics to a new, more powerful form of modern Materialism, became inevitable.

Later, when further work on the synaptic clefts of the brain showed it to be the scale (~ 300 nm) that Heisenberg's Uncertainty Principle could apply, it became feasible to apply modern Materialism to the brain itself. But I am getting a bit ahead of myself!

The point made here is that this newfound Materialism marked the first wholly PHYSICALIST version. This sort of Materialism allowed the concept of a "material thing" to be extended to include all of the elementary particles and other things that are postulated in fundamental physical theory, even continuous fields and points of space-time (see e.g. Richard Penrose's Twistor theory).

Up until the quantum theory, few thinkers had been able to discriminate the physical from the material. Most naturally assumed in simple-minded fashion that if a thing was physical it had to be material. Probably the first hints of this being false occurred in the 1850s with the electricity and magnetism experiments of Michael Faraday, who showed iron filings could be disturbed from a distance by an invisible force. That became known as "magnetism". Later - he placed filings on a card and let an electrical wire go through the card and still found the filings disturbed but now arranging themselves into circular patterns. Thus, he made the connection that a moving electric charge can create a magnetic field which has a similar effect to a magnet!   See e.g.


Moving on, one beholds a more refined form of modern Materialism which is usually referred to as Hylozoistic Materialism. In this paradigm, everything is composed of physical entities (generally) but it introduces special laws applying to complexes of physical entities, such as living cells or brains, that are not reducible to the laws that apply to the fundamental physical entities. (To avoid inconsistency, such a theory may have to allow that the ordinary laws of physics do not wholly apply within such complex entities.) Such a theory, could also be called "emergent Materialism," and these can shade off, however, into theories that one would not wish to call Materialist. These ascribe vital characteristics to all matter, while its counterpart, panpsychism, attributes a mindlike character to all constituents of material things.

A couple of examples include Rupert Sheldrake's "morphogenetic field" biology, which basically maintains that cells divide and organize based on a supra-physical field that carries some intentionality with it. British researcher Lyall Watson carried this even further in his controversial book, Supernature.

In general, however, modern Materialism disavows such "intentionality" as being unnecessary and moreover, harboring an unproven teleology. (All physicalist Materialisms reject teleology - or the imputation of hidden purpose - in any form) Up to now Sheldrake, for all his words, hasn't been able to isolate his "morphogenetic field" and neither has Lyall Watson shown that some "super mind' exists in nature and directs it.

Rather less drastic is epiphenomenal Materialism, according to which sensations and thoughts do exist in addition to material-physical processes but are nonetheless wholly dependent on material processes and without causal efficacy of their own. They are related to material things somewhat in the way that a man's shadow is related to the man. In this form, the mind is defined as a kind of immaterial "shadow" of the brain. As we know, if a man is eliminated so also is his shadow, and in like manner, when a brain dies (brain death, no EEG signals) there is no more operative mind. Thus, mind does exist but is contingent for its existence on the brain. Up to now no one has been able to prove that a human mind yet functions once the brain is deceased.

Less well known, physicalist Materialism groups into deterministic and indeterministic categories. In the former, we find people like Albert Einstein, who refused to accept the brain's thoughts were able to arise other than by predictable, deterministic interactions. However, with the advance of modern quantum theory this simplified view gave way to an indeterministic paradigm. Much of this was championed, for example, by quantum physicists such as Henry Stapp.

Stapp pointedly noted that uncertainty principle limitations applied to calcium ion capture near synapses shows they (calcium, Ca++ ions) must be represented by a probability function.[1] More specifically, the dimension of the associated calcium ion wavepacket scales many times larger than the calcium ion itself. This nullifies the use of classical trajectories or classical mechanics to trace the path of the ions.

However, no sooner had he explicated this, than a new form of quantum determinism arose from the Stochastic Interpretation of Quantum Mechanics used by David Bohm. This then provided a more refined vehicle for determinism to operate, but this time in a quantum acausal form. Much of this was described in an earlier blog post:

Inteestingly, both indeterministic and deterministic forms were found to be amenable to brain neural net manipulation, posing the possibility of remaking the human brain-mind. See:

http://brane-space.blogspot.com/2010/11/new-brain-new-mind.html

Interestingly, in a number of ways acausal determinist quantum theory has fed directly into another offshoot of Materialism known as Central state Materialism. On account of recent developments in biochemistry and in physiological psychology, the plausibility of Materialism overall has greatly increased. At the same time, there has been a resurgence of interest in the philosophical defense of central-state Materialism. Central-state Materialists have proposed their theories partly because of dissatisfaction with the analytical behaviorism of the Oxford philosopher Gilbert Ryle. Ryle himself is reluctant to call himself a Materialist, partly because of a dislike of all "isms" and partly because he thinks that the notion of matter has meaning only by contrast with that of mind, which he thinks to be an illegitimate sort of contrast. (In other words, a kind of false choice).

It is now generally accepted that Scientific Materialism as applied to the domain of natural science is in the physicalist form.  Hence Materialism as it is understood today embraces all physical fields and interactions, of both matter and energy. The laws that govern these interactions  (e.g,. the 2nd law of thermodynamics) apply without exception to humans as well as inanimate particles/objects. The prediction of the future behavior of all physical interactions is not dependent on the existence of any supernatural agency. Thus, physical laws are complete in the sense of being able to account for all physical phenomena.

Physical scientists then have at hand a modus operandi for understanding the universe which does not entail inclusion of supernatural agents, now regarded as redundant. Not surprisingly, the Materialist is much more likely than the supernaturalist to place a premium on revering the Earth and demanding the rational disposition of its resources. As a Materialist, after all, I can examine the evidence and determine that our planet is possibly the only inhabited one, at least in our galaxy - if not the cosmos. I can also ascertain that this life is most likely the only one and that I must strive to enhance it in any way possible. I should emphatically not squander what I have now, while awaiting a mythical afterlife.

I refrain from looking to any hypothesized deity for deliverance, or lay blame for human ills on some mythical demonic entity. On the contrary, Man alone is responsible for his actions and is the ultimate master of his fate. As a Materialist I maintain that Man need not suffer extinction as a species if he has the courage and vision to assume control of his destiny through the use of reason.
--------------------------------------
[1] Stapp, Henry, P.: 1993, Mind, Matter and Quantum Mechanics, Springer-Verlag, p. 42.

"Quarantine Fatigue" - The Latest Meme Hatched By FOX's Coronavirus Truthers? You Bet!


Numskulls & Shutdown "fatigued" wimps crowding Huntington Beach Sunday, even as the virus lurks among them - ready to roar into more overrun ERs. 
Even bigger crowd of numskulls and quarantine wimps at Newport Beach, CA, eager to prove Darwinian theory's culling of the dumbest and most reckless.
"The whole point of the shutdown was to buy ourselves time so we could do it right the second time after we muffed the first response. It's not only important to avoid opening up too aggressively but  if you don't have a good amount of testing, tracing and isolation, you won't be able to keep infected people away from susceptible people. The basic biology then is the infection is just going to take off again.  So unless the biology has changed we're going to end up in the same place and that is what we're all worried about."  - Dr. Ashish Jha,  Professor of Global Health, Harvard University, on 'All In' Monday night

"Despite Donald "Why Not Shoot Up Lysol?" Trump's unsubtle yearning to "reopen" the economy (which won't work) and let the novel coronavirus run rampant, a new poll from the Washington Post and the University of Maryland shows that strong majorities of Americans think that's a really stupid idea. Eight out of ten Americans took one look at the yahoos protesting the lockdowns in various state capitals and said, nah, living is more important than being able to order that 400-calorie vanilla-hazelnut frappuccino inside the Starbucks,:  -  Amanda Marcotte, smirkingchimp.com blog


Have  the last remaining sane and intelligent citizens finally been converted into morons by the raging Trumpie protesters?  That is what we initially considered upon watching the scenes of crowded beaches on the nightly (NBC) news yesterday.  An hour or so later, we were gratified -when Rachel Maddow announced on her MSNBC show: "The vast majority of Americans remain committed to shelter in place guidelines".   New polling appears to support this vote for common sense,, but one wonders how long it will last.   

Seeing the crawlers on the earlier newscast ("Quarantine Fatigue strikes Americans") was definitely not encouraging. to future confidence that citizens will remain in shutdown mode.   But then on further inspection it appears the term evidently originated within the rat warrens of the "Coronavirus Truthers" hunkered down at FOX News.  (See e.g.''Rightwing Media Spreading Epidemic of Misinformation' in one of the links at end of this post)
 Despite the FOX trolls, we ought to all, at least the higher IQ subset of the population, be able to agree on some general propositions:
1) The SARS- Cov-2 virus has spread as an unseen stealth agent among us, causing illness and death- the various forms of which (e.g. strokes in the young) are still not widely understood.
2) Despite the yen to reopen to business, at least 31 states   according to Harvard researchers--have still not met even minimal standards of testing.  These states will have to significantly step up testing to even mildly loosen 'stay at home' orders.  As Harvard Prof. Ashish Jha noted on All In last night, barely 2% testing is being done per day - which needs to be 10-20 times higher.
3) Given we have a vast "naive" population (with no immunity) and no vaccine, or treatment that works, it is pie-eyed foolish in the extreme to rush to reopen unless vastly more tests are run - up to 500,000 per day. Minus that testing, shutdowns are the only protection for a vulnerable populace (elderly, those with severe health issues).
None of this ought to be rocket science, so one is left to wonder why there is this "quarantine fatigue".  Are Americans such pathetic and entitled wimps an whiners that can't at least be content to remain locked down until June? For the sake of our hospitals, heroic health workers- many at the end of their tether?  Or must we again be reminded of Trauma nurse Amy Pacholk's words:
"All I keep hearing is that the curve is flattening, the curve is flattening. So that may be true but all the ICUs are still full. The sickest are still there. People are waiting to be intubated.  We're full we can't handle anything more. We can't open this country. If you want to live you stay home. My God,  don't open up this country. It needs to be closed until at least June."
Well, no, since polls show up to 8 in 10 Americans are not prepared to go out and have nails done, or eat out  - even if every burg in every state re-openend. The smart people know it is a fool's errand until the transmission rate is 1.0 or less. So, in the end we must deduce all those clogging beaches in the images are victims of FOX News and its grievance paradigm - or propaganda machine (also replicated in the WSJ's Op-ed pages)

Given we know the virus is a stealth assailant, it is highly plausible that amidst the tens of thousands  of  FOX-swallowers packing Florida and California beaches Sunday there may well have been lesser thousands of infected roaming amidst the unknowing masses. As an epidemiologist friend of ours at the Queen Elizabeth Hospital in Barbados put it:  "If any one of those beaches in the U.S. had 10,000 beach goers, you can be sure at least 1,000 were Covid-infected.  If the R nought is 3.0 that means each one of the infected could have spread the virus to 3 others, and each of those to three others in turn, i.e. 9 - and so on."
As I've written before in other posts, no one fancies an ongoing shutdown. Hell, I'd like nothing better than to get back to reading the papers over a sausage croissanwich, coffee and hash browns at  the nearest BK, but I can't.  Our Gov. Jared Polis has made it clear that even if some businesses can open seniors have to remain in lockdown mode for the time being. So no appearing at any packed venues, or inside dining. So, I am content just to pick up a bagel a few days a week at the nearby bagel place.
As the NBC News showed, tens of thousands of people packed southern California beaches over the weekend, reigniting fears that large crowds in public spaces could reverse progress on containing Covid-19 in the U.S.  Photos of the crowds in Newport Beach, Orange county, during the weekend  rightly sparked intense backlash and comparisons with Florida, where images of beachgoers raised alarms about the state’s coronavirus strategy. In recent days, beach and park reopenings have also prompted debates and public health concerns in Texas, Georgia, Mississippi, South Carolina and other regions looking to re-emerge from lockdowns.
But none of these impatient fools get that the virus is an unseen (stealth) killer and disabler) and it can be infecting the most healthy looking person. That person can spread it to you just by talking at close distance.  This is why California’s governor Gavin Newsom chastised those who crowded the beaches, saying “this virus doesn’t take the weekends off”.  No, and it doesn't pay any attention to human circulated memes like "the Covid can't take the heat!"
As Newsome went on to say:
This virus doesn’t go home because it’s a beautiful, sunny day along our coast,” 

The governor, who last week urged beach-goers to practice physical distancing, added at his daily news briefing yesterday:

We can’t see the images like we saw, particularly on Saturday in Newport Beach and elsewhere,”
As usual there is no unified plan.  Local officials in Los Angeles and the surrounding region have been deeply divided on the merits of allowing beach visitors. Many popular beaches have remained fully closed to the public due to Covid-19, though a handful have taken steps to permit visitors in a limited capacity while adopting a range of restrictions, enforcement plans and recommendations.
California’s statewide shelter-in-place order remains in effect, and police at the open Orange county beaches have attempted to enforce distancing rules. But there have been growing concerns over the weekend that the crowds were too dense, and the Newport Beach city council is now considering closing the beaches for the next three weekends or possibly blocking roads to the most popular destinations.
At Newport Beach, where temperatures were close to 90F (32 C), residents compared weekend crowds to the Fourth of July holiday, and lifeguards reminded people to stay apart if they were in groups of six or more. Police officials told the Los Angeles Times the department did not issue any citations for violations of stay-at-home orders.
Neighboring Huntington Beach in Orange county also saw big gatherings, despite the closure of parking lots along the highway. Sitting on a bench by the beach, Robin Ford surveyed the crush of visitors with concern.  Ford told the Orange County Register.:
Unless all these people are in one household, it does look like they are not social distancing,  
Meanwhile, Los Angeles city and county beaches, trails and playgrounds were closed, and officers on horseback were patrolling those areas to enforce social distancing rules. The city also opened cooling centers for people who might not be able to survive the heat wave at home, according to LA mayor, Eric Garcetti.
Barbara Ferrer, the head of the LA county health department, urged LA residents not to crowd the neighboring beaches that remain open.
Further north in California, police in Pacific Grove said they had to close the picturesque Lovers Point park and beach at the southern end of Monterey Bay due to lack of social distancing.
The question of beach access is likely to become more contentious as the U.S. heads into summer. Some experts have argued that governments should find ways to allow people to visit beaches while strongly encouraging distancing and other safety measures.

This isn't a remote issue in Barbados which has enforced a 24 hour lockdown now for weeks, with penalties of up to a year in prison and $50,000 fine for scofflaws.   See e.g.

Janice's take as usual is no nonsense on the comparison:  "Bajans have more sense than Americans and more self-discipline. They don't whine and cry and act like brats if they don't get their way.  They sacrifice for the greater good."

But not only Bajans.  75 years ago Americans also self-sacrificed with limited gas, food rationing and other curfew limits during World War II.  But then  our people were tougher, more resilient and not so entitled - i.e. made soft by too many conveniences and a social media environment that weakened brains.  They - like my parents - put on their big boy pants and toughed it out for YEARS, not just months.   Moreover, the sacrifice was uniform and universal across the nation, not just practiced in certain regions, states. Contrast that with the nonsense we behold now, including a moron president who actually believes injecting disinfectant like Lysol can be a treatment. (Or embedding a UV light into the body).  Compare Trump to FDR who was a real leader during the trying times of the Second World War and you have all you need to know about how and why this nation has gone off the path of being a beacon for the rest of the world.  And will likely cease to be the world's foremost superpower within the next ten years.

As of Monday, California has had 43,464 positive cases and 1,755 deaths..  Here in Colorado we have over 13,500 cases and nearly 600 deaths. Look for those numbers to explode as the morons  - such as shown in the images- keep having their tantrums about "quarantine fatigue". and claiming (as the Gridlock nuts are here in Colo., e.g. 'Inside State's Growing Shutdown Movement' Denver Post, p. 1A yesterday) they have a god-given right to "liberate" themselves.   Just take a gander at one of the quotes (ibid.) out of the mouth of a 'Gridlock'  protest leader  (and Coronavirus truther) named Joel Oltman from Castle Pines:

"If I choose the wrong way and it ends up killing me, I'm sorry but I made a choice.  I made an educated choice affecting me while respecting that other people deserve the right to feel safe."
Sounds all fine and noble, but he's totally oblivious to the fact that if  he's a stealth  (asymptomatic) carrier he can spread the virus to possibly hundreds of others.   How much "respect " is that for the rights of others to feel safe?  Of course, it goes without saying the guy is a Repuke.  As I said, all entitlement, no sense of sacrifice for the common good.  

See Also:



by Amanda Marcotte | April 28, 2020 - 7:15am | permalink
And:




And:




And:




And:
by Bill Berkowitz | April 29, 2020 - 5:39am | permalink

And:

by Bill Berkowitz | April 26, 2020 - 5:31am | permalink