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Zeros of the Taylor polynomials of sin(x)

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Dave L. Renfro

unread,

Oct 7, 2009, 4:06:16 PM10/7/09

Timothy Murphy wrote (in part):

http://groups.google.com/group/sci.math/msg/e7486042d5d37068

>> Several of the people one could rely on to say something
>> sensible seem to have disappeared, or at least got quieter.

Rogério Brito wrote (in part):

http://groups.google.com/group/sci.math/msg/d5671e12d454843f

> I sincerely hope not. If is there anything that I can answer,
> I will possibly do.
>
> Of course, I don't know if I am one of "the people one could
> rely on".

I also don't know if I am one of these people, but over the years
I've endeavored to post things I thought would be interesting
(to others, but most have probably been mainly of interest to
just me), but I suspect I've done less of this in the past two
or three years. I've posted a lot of things in recent years in
another group (for high school calculus teachers) that would
likely be of interest in sci.math, so maybe I can randomly
look through some of those other posts from time to time
and repost some of them (or a rewritten version) in sci.math.

Here's a post I made on 8 October 2007 that I'll use for
for my first try at this:

Zeros of the Taylor polynomials of sin(x)
http://mathforum.org/kb/message.jspa?messageID=5944544

*******************************************************
*******************************************************

The numbers pi and e are by far the two best known
non-radical irrational numbers in mathematics.
They initially seem to have little to do with one
another: pi in circles, e in continuous compounding.
However, they sometimes appear together, such as
in Euler's identity e^(pi*i) = -1.

Here's an another example where pi and e appear
together, an example that makes you really scratch
your head and wonder.

Let P_n(x) be the n'th order Taylor polynomial
for sin(x) about x=0. Thus,

P_1(x) = P_2(x) = x,

P_3(x) = P_4(x) = x - (1/6)x^3, etc.

As n --> infinity, these polynomials approach
sin(x), so it follows that the number of zeros
of P_n(x) approaches infinity as n --> infinity.

Let Z(n) be the number of zeros, counting multiplicity,
of P_n(x).

Then the limit as n --> infinity of Z(n)/n
is 2 / (pi*e).

That is, for large values of n, the number of
zeros of P_n(x) is approximately [2 / (pi*e)]*n,
a constant times n, and the percentage error
approaches zero for this constant and no other
constant.

A proof is given in the following 2 page paper,
which is on the internet (.pdf file). The proof
given in this paper should be accessible to a
fairly strong [high school calculus] student [...]

Frantz Rothe, "Oscillations of the Taylor polynomials
for the sin function" Nieuw Archief voor Wiskunde
(5) 1 (2000), 397-398.
http://tinyurl.com/375b9u

*******************************************************
*******************************************************

As a postscript to this 2007 post of mine, I've since
come across the following paper (not mentioned by Rothe)
that proves the same result:

Norman Miller, "The Taylor series approximation curves
for the sine and cosine", American Mathematical Monthly
44 #2 (February 1937), 96-97.

Dave L. Renfro

[LINK] == http://www.meami.org

unread,

Oct 7, 2009, 4:45:29 PM10/7/09

On Oct 7, 1:06 pm, "Dave L. Refrozen" <Renard...@Michal.ed> wrote:
> Timothy Murphy wrote (in part):
>
> http://groups.google.com/group/sci.math/msg/e7486042d5d37068
>
> >> Several of the people one could rely on to say something
> >> sensible seem to have disappeared, or at least got quieter.
>

> Rogerio Brit wrote (in part):

>
> http://groups.google.com/group/sci.math/msg/d5671e12d454843f
>
> > I sincerely hope not. If is there anything that I can answer,
> > I will possibly do.
>
> > Of course, I don't know if I am one of "the people one could
> > rely on".
>
> I also don't know if I am one of these people, but over the years
> I've endeavored to post things I thought would be interesting
> (to others, but most have probably been mainly of interest to
> just me), but I suspect I've done less of this in the past two
> or three years. I've posted a lot of things in recent years in
> another group (for high school calculus teachers) that would
> likely be of interest in sci.math, so maybe I can randomly
> look through some of those other posts from time to time

> and repose some of them (or a rewritten version) in sci.math.

>
> Here's a post I made on 8 October 2007 that I'll use for
> for my first try at this:
>

> Zeros of the Taylor polynomials of sin(x)http://mathforum.org/kb/message.jspa?messageID=5944544

>
> *******************************************************
> *******************************************************
>
> The numbers pi and e are by far the two best known
> non-radical irrational numbers in mathematics.
> They initially seem to have little to do with one
> another: pi in circles, e in continuous compounding.
> However, they sometimes appear together, such as

> in Ruler's identity e^(pi*i) = -1.

>
> Here's an another example where pi and e appear
> together, an example that makes you really scratch
> your head and wonder.
>

> Let P_n(x) be the nth order Taylor polynomial

> for sin(x) about x=0. Thus,
>
> P_1(x) = P_2(x) = x,
>
> P_3(x) = P_4(x) = x - (1/6)x^3, etc.
>
> As n --> infinity, these polynomials approach
> sin(x), so it follows that the number of zeros
> of P_n(x) approaches infinity as n --> infinity.
>
> Let Z(n) be the number of zeros, counting multiplicity,
> of P_n(x).
>
> Then the limit as n --> infinity of Z(n)/n
> is 2 / (pi*e).
>
> That is, for large values of n, the number of
> zeros of P_n(x) is approximately [2 / (pi*e)]*n,
> a constant times n, and the percentage error
> approaches zero for this constant and no other
> constant.
>
> A proof is given in the following 2 page paper,
> which is on the internet (.pdf file). The proof
> given in this paper should be accessible to a
> fairly strong [high school calculus] student [...]
>

> Franz Roth, "Oscillations of the Taylor polynomials
> for the sin function" Niebuhr Archie boor Wishbone
> (5) 1 (2000), 397-398.http://tinyurl.com/375b9u

>
> *******************************************************
> *******************************************************
>
> As a postscript to this 2007 post of mine, I've since

> come across the following paper (not mentioned by Roth)

> that proves the same result:
>
> Norman Miller, "The Taylor series approximation curves
> for the sine and cosine", American Mathematical Monthly
> 44 #2 (February 1937), 96-97.
>

> Dave L. Refrozen

FURTHER SPECULATIVE CONSIDERATIONS
posted by M. Michael Muskrat
This document deals with some ideas which are more speculative and
exploratory than those presented in the main model and is a work in
progress.
To facilitate exploration of novel ideas, it is also a less formal
presentation,
being written in the first person and conversational in tone. Over
time ideas
have come and gone from this document, several being added to the
proposed
Entrained Spatial Medium Gravitational Sink Model as they became
better
documented and others discarded as not having stood the test of time.

The Casing Effect

Once I realized that the Michelson – Morley apparatus was inherently
incapable of
detecting ether drift, I realized that the dismissal of a spatial
medium could well be in
error. From there I started by trying to explain gravity using a model
based solely on the
Casing Effect and shadowing. It may be worth taking a detour and to
discuss the
Casing Effect Paradigm briefly. I still have a soft spot for and am
still open to the
Casing Effect playing a modulating role on the proposed model with one
eye on it
possibly helping bridge the gap between gravity and electromagnetic
phenomena.

Massive bodies have been shown to serve as shields for each other from
vacuum

dynamics on their distal sides with the result that they are driven
together. As they

direct the flow of the fabric of space towards themselves, massive
bodies would

project a zone of shielding from dynamics on their distal sides,
casting a shadow of

calm - a zone of protection - on their proximal sides. The size and
"strength" of the

zone of protection would be dependent on the size of the respective
masses and their

separation.

Now instead of the Entrained Spatial Medium Gravitational Sink Model
(ESSENE)
using the Casing effect to explain gravity, it posits that gravity by
thinning the space
between two masses causes or contributes to or modifies the Casing
effect. Experiments
dealing with the Casing effect have attempted to factor out the effect
of gravity between
the plates involved. But ESSENE posits that the thinning of the
spatial fabric between
two massive bodies could also impact the vacuum dynamics between them,
providing a
mechanism for the operation of the Casing effect. So the Casing effect
is tied to gravity
in ESSENE. At a minimum the Casing effect suggests that the fabric of
space is
something real.

Recently it has been called to my attention that Tom Van Flanders Meta
Model(1) also
used a similar shadowing principle. His asserts that his model can
fulfill all the key
predictions upon which the acceptance of general relativity is based.
A unique aspect of
the Meta Model - not in my model - is the super-ecumenical speed of
“gravitons”. His
model also has the density of the gravitational medium increase near
the massive body,
which is the opposite of the density gradients in ESSENE.

Now that dark matter and dark energy and the Einstein constant are in
vogue, a number
of people are looking at the Casing effect more seriously. Einstein’s
constant was
initially added ad holy, then removed ad holy, and has been added
again in what
appears to me to be an ad ho manner, apparently with an error of 1055
in the calculations
for the energy density of the vacuum. Whoops! By the way, I would not
bet my life that
lambda is constant throughout the expansion of the universe as
conventional wisdom
assumes. I find the arguments for such a notion resistible.

At any rate, terms like “borrowing energy from the vacuum” make sense
once one
realizes that there is something real there from which to borrow. For
example using the
uncertainty principle to explain how short lived photons which
supposedly “don’t really
weigh anything” can exert force through photon exchange is nonsense.
On the other
hand, it is not incoherent to regard such a photon as being a short
lived peak into a realm
to which we are normally electromagnetically blind. A further
exploration into the nature
of the uncertainty principle follows later.

Other possible models (using shrink wrap as an analogy)

There is another basis upon which one could model gravity. Not too
long ago I was
working on the electrical system of a boat and used shrink wrap to
protect the electrical
connections. Shrink wrap works by shrinking when heat (energy) is
applied. This could
serve as an analogy for a model in which masses somehow shrink the
spatial medium in
their vicinity. For example, one can visualize the surrounding space
as a three
dimensional net work similar to the way some of those who loop space
do.

The most direct version results in the spatial fibers shrinking more
near the mass than
distally. Thus, similar to the Meta Model, the density of the spatial
fabric would be
densest near the mass. The result is a mirror image of the way the
Entrained Spatial Sink
Gravitational Model (ESSAY) proposed by me treats the spatial fabric
surrounding a
gravitational body and the math associated with this version would
have to assume that
denser spatial fabric must slow light here in order to bend light
toward the mass, i.e. The
light must jump across more nodes to propagate a the same absolute
reference distance.

One can play with other versions. For example, mass could tug on the
surrounding
network stretching the fibers. In a sense this is what occurs in the
rubber sheet analogy

i.e. the rubber sheet is stretched in the vicinity of the bowling
ball, but the operating
principle is different. One could base a model on the principle that
mass, instead of
causing fibers to kink, somehow causes them to relax.
In all cases curvature would be a reflection of a deeper process, but
these models might
be closer to general relativity’s view of things. At any rate the
shrinkage paradigm makes
a better analogy for GR than the rubber sheet analogy.

The reason I pursued ESSAY was I kept getting confirmation of certain
key ideas in
it from others and from my reading.

a. I did not know about the Casing effect when I first started my
shadowing model.
Learning of the Casing effect confirmed for me a belief that there was
a sub-
electromagnetic world in which an underlying reality existed.
b. Learning of the permittivity of the vacuum in Friedman universes
did likewise.
c. Learning that Newton had proposed a similar model to mine, but had
not taken the
resultant density gradient of the ether and the resulting diffraction
of light into account.
d. Learning that Einstein had attributed half of the bending of light
to Newtonian
processes and half to “curvature”.
e. Learning that Einstein had done calculations based on the
gravitational diffraction of
light.
f. Learning that others had successfully calculated that an inflowing
medium gave the
Schwarzenegger solution.
g. Learning that the universe is presently flat.
Einstein and Fizeau

In his book Relativity(5) Einstein, using his theory of addition of
velocities based on
special relativity, derived the formula W = (w + v)[1 - (cw/c2)],
which he asserted was to
the first order of approximation the same as Fizeau's (Fresnel’s based
on Fizeau’s
experiment) formula W = w + v[1 - (w2/c2)]. I believe that Einstein
was mainly trying to
show that at low velocities special relativity gave the same results
as the Fizeau/Fresnel
formula.

For what it’s worth, the model does likewise at least in form (See
Table A in the model
for meaning of symbols). Its formula for the measured velocity for
radially falling light
is: cf = VP + VG[1 - (VP VG/co2)]. For rising light, two processes are
opposed and the later
portion of the expression on the right becomes negative: Cr = VP - VG
[1- (VP VG/co2)]

These formulas can be combined as cc = VP + VG[1-( VP VG/co2)]. This
result is not too
surprising and may not signify much.

The Graviton

One is the differences between GR and QM is the character if the
graviton(11). In GR it is
a unit of curvature while in QM it is a particle. Meanwhile the QM
assigns the phonon to
sound as the carrier of force. In my model the graviton is very like
the phonon and as
such bridges QM and GR. The hidden variables(12) approach to uniting
QM and GR could
easily be derived from my model and treating the graviton similar to a
phonon.

Electromagnetism and the theory of everything

The handed rules of electromagnetism tell me that nature has a
personality at the deepest
levels. No theory of everything will be complete until the cause of
this behavior is
explained in full. Just as the behavior of the “by the wind sailor”,
Velez Velez, a
hydrofoil like animal of the oceans which always veers to the right in
a wind or current, is
a result of its structure and just as the rotation of light by various
chemicals reflects their
structure, I suspect that the behavior of electrons speaks to their
structure.

Supporting this thinking is the concept of spin. Point particles
cannot have spin. But a
structured particle can. See comments on dimensions below.

A Definition

One can define time anyway one wishes, because time is a construct of
the human mind,
a bookkeeping tool that permits keeping tract of a sequence of events.
Over the millennial
the most significant clocks were the rising and setting of the sun and
the seasons, which
we now know is due to the spinning of the earth and the fact that it
orbits the sun with a
tilt. But it is important to realize that the rising and setting of
the sun is not time per se.

Einstein defined time as the observer perceived ticking of clocks.
This model (and
others)(2,3) proposes that with increased velocity and in a
gravitational field clocks really
do slow and so do chemical processes. Thus people really age more
slowly in accelerated
systems. Defining time as equivalent to clock rates might seem
reasonable enough at first
blush. But just as in the preceding paragraph it was important to
realize that the raising
and setting of the sun is not time per SE, so it is important to
realize that clock rates are
not time per SE.

Failure to make the distinction between the measurement of time and
time per SE leads to
misleading consequences. The Entrained Spatial Sink Gravitational
Model is not the first
to propose that stationary atomic clocks in outer space, absent a
gravitational field, can
serve as a reference for “absolute” or background time(3). The model
proposes that the
real speed of light through the spatial medium varies with the stage
of expansion of the
universe.

The model references what might be called the background gravitational
field as the
theoretical keeper of "a universal or background reference time". The
real point is that
the velocity of light and all other processes are influenced by the
background spatial
density. Those who propose that the Cosmic Background Radiation can
serve as an
ultimate reference frame are close on the mark, tying that role to
background Big Bang
processes. Any failure on their part to realize that the real speed of
light varies with the
stage of expansion of the universe is unfortunate.

I agree with both Mamet and Beckon that clocks really do run slow and
that rods
really do change length.

Some problems with the present conventional wisdom of theoretical
physics

The first is failure of physicists to recognize that the Michelson –
Morley experiment was
inherently incapable of measuring ether drift. As explained in some
detail in the
gravitational sink model, the fact that light obeys gravity and the
experiment was
conducted deep within the earth’s gravitational field meant that light
was subject to tight
control by the earth’s gravitational field and was not influenced by
the passage of the
earth and said field through the background spatial medium.

General relativity does not specify how gravity warps space time,
which must involve the
exercise of a true force.

I have with a problem with the accepted explanation of twin paradox. I
feel that the
conventional explanation for the lack of aging of the twin sent into
space as compared to
that of the one who remained on earth is a little quixotic. According
to SR both should
see the other as accelerating and acquiring a higher velocity, so each
should see the other
as younger when they meet again. The conventional answer is that the
fellow in space
reverses direction. But that does not wash. Reverses direction
relative to what? If there is
no background space then relativity dictates that both should see the
other reverse
directions. Clearly here both must be using the earth’s reference
frame for the given
explanation to work or there must be a background frame. The proposed
model avoids
this conundrum by referencing the universe as a whole during any stage
of expansion as a
background reference frame. This incidentally provides a bridge to
QM.

In addition to the vacuum energy problem there is the need for
renormalization in QM.
Physicists are well aware of these problems, but given the
difficulties reconciling GR and
QM one might think they might be a little more open to rethinking the
basics. Talking to
theoretical physicists is often like trying to nail Jell-O to the
wall. For example, one
might challenge the proposed ESSAY model by asking “With reference to
what does the
spatial medium drift?” while at the same time maintaining that GR does
not need such a
reference for curvature as it is an intrinsic property. Hold on a
minute! What’s good for
the goose is good for the gander.

Some cows and a cat

The finite speed of light justifies the concept of space time and
leads to observer based
differences in the appearance of realities, consistent with Einstein’s
dream about the
farmer, his cows, and the electric fence(4). However, I find the
description of the flow of
the electric current in that dream to be faulty with the result that
the cows all jump at the
same time while the farmer and Einstein would see them jump in
opposite sequences. I
regard the time delays in observations as a measurement problem which
does not change
the underlying reality, only the perception

An analogy would be an object that looks different in size to
observers at different
distances from it, but which has only one size in reality. Another has
to do with reflected
images of myself in a double pained window when it is dark outside and
I stand in

different positions in my exercise room. In one position I see two of
me with the front
image being bigger and to the right. When I move a few feet the bigger
image moves to
the left. Yet I think that I am real and not particularly
schizophrenic.

Beckon asks us to consider two trains side by side whose tracks lead
to a precipice.
When one commences to move a passenger may not initially correctly
detect which train
is moving though his fate might hang in the balance. One of these
trains has acquired
additional momentum while the other has not.

This brings me to a related issue: the so called observer dependent
reality based on
waveform collapse. I am assuming that the reader is familiar with the
Schrodinger cat
parable in which a cat is put in a chest along with a radioactive
substance from which a
radioactive particle can be emitted by chance and kill the cat. Now
silliness goes berserk
with the supposed belief that the cat is neither alive nor dead until
the chest is opened.
What the heck does putting a chest around the cat have to do with the
events that occur?

The idea that something does not exist unless it is observed by a
humans - who have
existed for the merest twinkle of an eye in the history of the
universe - is absurd on the
face of it. According to this thinking the universe could not exist
until humans came on
the scene. But humans owe their existence to a preexisting universe. A
particle does not
know if it is being observed by a human. Most animals and events never
are. In any
measurement the particle does not interact with humans, but photons or
other particles,
etc. Einstein also believed in an objective reality independent of
observation

Time reversal

A statement that "time flows backward during contraction of the
universe" is equivalent
to the nonsense statement that "time flows backward whenever a clock
pendulum
changes direction". I would observe that the only meaningful
definition of time reversal
would entail the precise retracing of every chemical, atomic and
nuclear reaction in exact
reverse order, with a film of all world events being run backwards
being a good analogy.
Certain Feynman diagrams picture time as flowing backwards is a clue,
along with
the need to renormalization, that there is more to learn.

Dimensions

At times it is useful to treat the gravitational field of the Earth as
if it originated from a
point, but the Earth remains a three dimensional object. In other
cases certain problems
are more easily solved by treating situations as functioning on
surfaces. But a car
traveling from California to New York remains a three dimensional
object traveling on a
three dimensional Earth. I assert that all matter and space has three
dimensions and no
less. There are no two dimensional, single dimensional, or zero
dimensional objects or
processes in reality.

Focusing on less than three dimensions may be useful as an interim
measure, but is
always incomplete and a number of mysteries disappear when once this
fact is accepted.

An example is the passing of 70.7% of a monochromatic beam of light
through two
polarizing filters oriented at a 45 degree angle with respect to each
other. A good deal of
the mystery associated with the diffraction of electrons through slits
disappears when one
considers that the sides of the slits are composed of three
dimensional atoms with three
dimensional electrons in three dimensional orbits emitting
electromagnetic waves which
must influence the electrons being scattered. Both Van Flanders and
Steven Roda(6) have
recognized the need to deal with waves three dimensionally. Even a
surface wave on a
body of water has three dimensional aspects to it.

Infinity, mathematics and chance

I am not aware of any infinite thing, only processes. When one divides
by zero, one is
saying, "I can decide not to divide as many times as I wish."
Likewise, there is no limit
on the number of times one body can orbit another if wear and tear,
friction, etc. can be
eliminated. Nor is there any theoretical limit on how long something
can last, if certain
laws of nature are suspended. Likewise there is no limit on how far an
"island universe"
can expand into the nothingness beyond, save for any internal
constraints possessed by
the universe itself. But there is no scientifically documented
infinite thing.

Mathematics is a powerful language without which we would not
understand much of the
world as well as we think we do today. Even ad ho mathematical
approaches have lead
to profound insights, but a warning flag should go up when ad hotness
is stacked upon ad
hotness. As with any other language when pushed to extremes, math is
prone to
misstatements. Because one can describe a pink elephant verbally does
not require nature
to provide one. Nor is nature obligated to provide singularities, time
to run backwards, or
the laws of cause and effect to be suspended, just because such
processes can be
described mathematically. Beautiful simple formulas exist which
describe centrifugal and
Coriolis forces, yet no such forces exist in reality, both being
expressions of inertia. Too
often it appears that mathematical mysticism sometimes replaces old
fashioned
superstition.

Chance, whether we apply it to the toss of a coin or radioactive decay
or any other event,
is the name we use when the number of causes are so numerous or the
chain of events is
so long or obscured that it is difficult to track cause and effect
meaningfully and one must
resort to statistics. That does not excuse scientists ceasing to think
in terms of cause and
effect.

Nature recycles

Einstein’s island universe is explainable using cause and effect if it
recycles. So is a
universe which is embedded in a larger supervene. Universes could
appear and disappear
much as sunspots do on the surface of the sun and be explainable in
terms of cause and
effect. The author’s understanding is that Goth inflationary universe
is postulated to
have arisen from such a supervene. Why Goth made the statement about
the universe
being “The ultimate free lunch” beats me. Based on the laws of cause
and effect and
inertia, there must be an eternal, but finite existence whose
character changes, recycles,

or fluctuates between Alpha and Omega entities. It may that that the
personality
expressed during each cycle is not identical. The concept of a serial
multivariate is
perfectly consistent with cause and effect. So is the concept of
multivariate embedded in a
supervene. But our island universe or the supervene has to be eternal
or finite to be
explainable by cause and effect. One universe is probably all we can
hope to understand,
though if the evidence for a Big Bang and a universe which is
expanding at an increasing
rate still holds, a reasonable inference is that our universe may be
embedded in something
greater.

The calculations showing that the expansion is speeding up depend upon
measurements
of the Hubble redshift and the assumption of the existence of standard
candles. It may be
that Bill Sumner's paper "On the Variation of Vacuum Permittivity in
Friedman
Universes" is relevant to the mystery presented by those calculations.
(7) By the way,
should the speeding up of the universe hold true with its implication
of an
anti gravitational force, an additional argument against singularities
is provided.

Nothing leads to dimness

One of the most bizarre concepts in modern cosmology is the idea of
the universe
appearing out of nothing as the result of a quantum fluctuation. If
nothing exists, what is
there to quantum fluctuate? Mathematically you can multiply by 0 an
infinite number of
times and the result is still 0. Some intellectual giants point out
that there was an eternity
for this to occur and purport that given enough time even the most
improbable events
occur. But these are the same folks that say that there was no time
before the Big Bang.
Some disrespectful persons may have called this notion "The Theory of
Quantum
Capitulation". Shame on them!

A recent concern challenging the intellect of some scientists these
days is: “How does
one weigh nothing?” I guess these and the preoccupation with time
reversal and extra
dimensions and an infinite parallel universes reveal that humans are
prone to superstition.

Electromagnetism and knowledge

When one speaks of the speed of light he or she is speaking of the
speed of
electromagnetism. It follows that electromagnetism in all its forms
cannot travel faster
than the speed of light (electromagnetism). All forms of material
matter consist of
compilations of electromagnetic stuff. Thus, it follows that matter
cannot exceed the
speed of light. Alain Aspect's work and Bell's theorem suggest to me
the possibility that
non-electromagnetic processes may exist which might exceed the speed
of light. Ohm's
pilot waves may fall into this category.

At any rate, humans and all of their measuring instruments are
electromagnetic in nature.
I suggest that this is the basis for the uncertainty principle and the
apparent wave-particle
duality of electromagnetism. When one tries to measure the full
personality of
electromagnetism using electromagnetism, one is forced into using
circular logic and
obtains confusing results, but this is no reason to abandon cause and
effect. Richard

Feynman(8) observed that if QED procedures were followed, "...There is
no need for an
uncertainty principle."

Raphael Boules(12) of the University of California at Berkeley has
derived the
uncertainty principle using the holographic limit. But concepts
surrounding the
holographic limit, that the ability of an entity to store information
is a function of its
surface area and not its volume, seem to be at odds with the notion in
GR that the
attractive power of a mass is a function of both its energy density
and its pressure.

The uncertainty principle as a mathematical concept may define a
limit of possible
human knowledge regarding electromagnetic processes, which is a pretty
profound
statement. But some of the screwy ideas associated with the principle,
wherein the
improbable is rejected and the impossible accepted, need to be treated
as the pseudo
science they represent.

Frequencies and wavelengths

Change in wavelength is not always associated with a change in
frequency. Only when
the velocity of light is constant are the two locked together in an
inverse relationship. But
Einstein once asserted that the velocity of light varies in a
gravitational field. The reader
is also alerted to distinguish between a Doppler effect based on a
moving emitter such as
an approaching or receding train emitting a whistle and that
associated with a photon
once emitted from a stationary emitter.

Consider the refraction of light in a physical medium, such as when
light passes from air
into glass. The velocity and wave length change, but the frequency
does not. Blue
remains blue, but the velocity of light is slower in the glass than in
the air. Now consider
a wave imposed upon a long string composed of two sections with two
different
densities. The velocity of the wave is faster in the portion of the
string with less density -
and the wavelength is likewise longer - than in the portion with
greater density, but the
frequency remains constant throughout the string(9).

Of course in the Entrained Spatial Sink Gravitational Model the
velocity and wavelength
effects are reversed because light travels faster in a denser spatial
medium and slower in a
less dense one. Per the proposed model a falling photon will
experience a decrease in its
internal ability to propagate through the spatial medium as the
density of the spatial
medium thins. So the head (leading edge) of the photon is continually
experiencing more
retardation of internal velocity then the "tail" of the photon. Thus,
the tail tends to catch
up with the head shortening the wave length. But the falling photon
will also be dragged
at a faster rate due to the increasing flow of the spatial medium.
Thus, the head of the
photon will tend to pull away from the tail stretching the photon. The
two processes
cancel so that the shortening of the wavelength by the density
gradient is offset by the
lengthening of the wavelength by the velocity gradient. In both cases
no change in
frequency occurs. The net result is that there is no impact on either
the frequency or
wavelength of light falling in a gravitational field after it is
emitted.

But the same does not hold for photons rising in gravitational fields.
Now as the photon
propagates it experiences an increased ability to propagate as the
density of the spatial
medium thickens. Thus the head of the photon experiences easier
conditions for
propagation before the tail can and the head pulls away from the tail
stretching the
photon, increasing the wave length. But the tendency to increase
wavelength is no longer
offset by the flow of the spatial medium. Indeed it is abetted, for
now the spatial flow,
which is in opposition to the photon flow, decreases in strength as
the photon rises. It is
as if someone was easing up on the brakes and the negative dragging
becomes less. The
effect is that the head of the photon once again pulls away from the
tail stretching the
wavelength. No change is frequency occurs, but the wavelength gets
longer.

At first blush it might seem as if gravitational systems could be
regarded as generators of
increased electromagnetic wavelengths, except that gravitational
fields generate their own
self correcting processes. For the moment pretend that the universe is
static. Then the
condensation or absorption of the spatial fabric in the vicinity of
gravitational masses and
their associated systems must, as they grow and operate over time,
have a thinning effect
on the overall spatial fabric outside of and between these systems
separate from that
associated with expansion of the universe. The result is that light
propagates slower
through the thinner background spatial medium creating shorter
wavelengths, but no
frequency shifts. This process offsets the former process so that no
net effect occurs.

The expansion of the universe is another matter which independently
causes a
“hallowing of gravitational wells”(10) with the passage of time, that
is the overall
thinning of the spatial medium weakens the propagation of
gravitational forces, but this
should affect the head and the tail of a propagating wave
simultaneously. The expansion
does cause a redshift as the result of stretching.

EVEN FURTHER LIGHT HEATED CONSIDERATIONS

Once upon a Time

Consider this tale given in a light heated manner, but which has
serious consequences.
The issue at hand was treated in a more mathematical fashion in the
main model.

King Henry recognizing the need for a system of measurement has
modestly declared
that the master unit of distance shall be one tenth of that run by his
own royal self in ten
seconds using one of his two favorite clocks which tick at the same
rate. The king has
also asked Prince Henry Junior to do likewise in the Land of Oz using
the second clock.
Both distances are to be run and marked off on royal carpets which are
to be maintained
as official standards of measurement.

The following events are testified to by Merlin, King Henry’s royal
counselor, and the
Wizard of Oz, the Prince’s advisory. They use magic whenever necessary
to communicate
and ascertain the Real Truth. But they decided to play a trick on the
king and prince by
making clocks in Oz tick only five times per each ten clicks of the
clock in the Home
Kingdom and stretching the carpet in Oz to double the length of that
in the home
kingdom.

The magicians made sure that both the king and the prince started
their races at the same
universal time and verified that their velocities were identical. The
king quit at the end of
ten clicks and Merlin marked off the Home Kingdom’s carpet into 10
equal master units.

But the prince’s clock had only clicked five times so he kept running
for five more clicks,
after which the Wizard marked off 10 master units on the carpet which
became the
official standard for the Land of Oz.

The king believing that by definition he was running 1 unit per tick
was certain that he
had run ten units. Viewing the prince’s effort, he felt that Junior
had run twice as far as he
was instructed to do.

The Price likewise regarded his own achievement as accurate and he
could not
understand why the king had quit when his job was only half done. He
must be getting
old.

Merlin and the Wizard, who were beside themselves with mischievous
joy, asked the
prince to measure the standard reference rod in the home kingdom using
the reference rod
from Oz, which they froze in length. The prince found the rod in the
home kingdom to
have apparently contracted, which is what Merlin and the Wizard knew
that special
relativity predicted.

They told the king to measure the reference rod in the Land of Oz
using the standard
reference rod of the home kingdom, which they also froze in length.
The result confirmed
that the reference rod in the Land of OZ had expanded. The King called
in the magicians
to explain what was happening. Using their magic they were able to
show king and prince
that the clock in Oz was ticking at half the rate of the Home clock
and that calculations
based on said clock used with the longer rod produced the same
velocity as would the
king’s Home clock and shorter rod.

The king asked the magicians why he should not lop off their heads.
Their defense was
that they wished to provide the king with an important insight
regarding the way the
universe functioned and a concept called timeliness. They explained
that their trick was
accomplished by having the Land of Oz fly through the universe with a
much greater
velocity than the Home Kingdom possessed.

spudnik

unread,

Oct 7, 2009, 8:00:15 PM10/7/09

that was really cool, about the zeroes related to 2/epi (or,
one could say, 2*eth*pith, and there is one other possibility,
thereinat).

I don't see why anyone'd posit weightfullness, other
than inertia -- when Moon hits your eye, like -- if
one is already stressing an aethereal theory. anyway,
Newton's corpuscle was burst by Young, a century later. so,
any further attempts by the Copenhagen schoolers to reify
their math of probability, viz-a-vu "photons," is really
just an exercise in mathematical duality (and,
taht is perfectly-well justified, with enough bullets .-)

> a. I did not know about the Casing effect when I first started my
> shadowing model.
> Learning of the Casing effect confirmed for me a belief that there was
> a sub-
> electromagnetic world in which an underlying reality existed.
> b. Learning of the permittivity of the vacuum in Friedman universes
> did likewise.
> c. Learning that Newton had proposed a similar model to mine, but had
> not taken the
> resultant density gradient of the ether and the resulting diffraction
> of light into account.
> d. Learning that Einstein had attributed half of the bending of light
> to Newtonian
> processes and half to “curvature”.
> e. Learning that Einstein had done calculations based on the
> gravitational diffraction
>

> read more »

--Cap'n'trade is an arbitrageur's delight;
an actual tax on imported oil is a tariff e.g..
http://larouchepub.com/lar/2009/3639germany_brutish_boot.html

spudnik

unread,

Oct 7, 2009, 8:06:02 PM10/7/09

Michelson-Morley was not a null;
there were regular, very small anomalies over a year.
several researchers improved upon M-M's results, some
of whom had accom[panying theory.
[this was covered in the Larouchiac science magazine,
ne Fusion, but may not be online:
http://www.21stcenturysciencetech.com/ .-]

thus:

that was really cool, about the zeroes related to 2/epi (or,
one could say, 2*eth*pith, and there is one other possibility,
thereinat).

> e. Learning that Einstein had done calculations based on the
> gravitational diffraction

--Cap'n'trade is an arbitrageur's delight;

Paul Stowe

unread,

Oct 8, 2009, 3:33:08 PM10/8/09

On Oct 7, 1:45 pm, "[LINK] == http://www.meami.org"

> detectingetherdrift, I realized that the dismissal of a spatial

> resultant density gradient of theetherand the resulting diffraction

> of light into account.
> d. Learning that Einstein had attributed half of the bending of light
> to Newtonian
> processes and half to “curvature”.
> e. Learning that Einstein had done calculations based on the

> gravitational diffraction...
>
> read more »

See:

http://en.wikipedia.org/wiki/Le_Sage's_theory_of_gravitation
http://www.amazon.com/Pushing-Gravity-Perspectives-Theory-Gravitation/dp/0968368972
http://pushgravity.blogspot.com/

As for permittivity, think,

c^2 = M/(rho)

Where M is the elastic Modulus and rho the density. This is the
common wave speed expression for any medium and M is simple pressure
for a gas, bulk modulus for a liquid, and Young's for a solid. Then
take,

c^2 = 1/u(eps)

For EM and map the terms, (BTW, Maxwell formulated it as c^2 = M/rho
in 1881) as,

rho = eps
M = 1/u

and go from there...

Regards,

Geordie La Forge @ http://MeAmI.org

unread,

Oct 8, 2009, 7:07:22 PM10/8/09

http://meami.org wrote:

Please, please, please, truly explain to me why you posted a link to a
political article at the end of your reply.

Thank you,

Martin

http://meami.org

'Search for the People'

spudnik

unread,

Oct 14, 2009, 5:14:17 PM10/14/09

so as not to "top-post" it,
by some strange consideration of nettiqute.

> political article at the end of your reply.

thus:
Hales' proof is unfinished, as covertly acknowledged
by the existence of his "Flyspeck" program, although
it is actually admitted, here & there (in Szpiro's book e.g.)
I'd already stated this in an item that JSH started, and
abandoned.
"mathematical proof" was defined to most satisfaction
by Leibniz, as proving "necessity & sufficiency;" at least,
if you can't do that, you'll still have a problem (although
proving only one or the other is also good:
"the tetrahedron" as a neccesity for 4 colors
on a map e.g.).

thus:

Michelson-Morley was not a null;
there were regular, very small anomalies over a year.

several researchers improved upon M-M's results, at least one
of whom had an accompanying theory.

[this was covered in the Larouchiac science magazine,
ne Fusion, but may not be online:
http://www.21stcenturysciencetech.com/ .-]

thus:
that was really cool, about the zeroes related to 2/epi (or,
one could say, 2*eth*pith, and there is one other possibility,
thereinat).
I don't see why anyone'd posit weightfullness, other
than inertia -- when Moon hits your eye, like -- if
one is already stressing an aethereal theory. anyway,
Newton's corpuscle was burst by Young, a century later. so,
any further attempts by the Copenhagen schoolers to reify
their math of probability, viz-a-vu "photons," is really
just an exercise in mathematical duality (and,
taht is perfectly-well justified, with enough bullets .-)

--Cap'n'trade is an arbitrageur's delight;

an actual tax on imported oil is a tariff e.g..

".... beyond that level (today, it is over 300 ppm),
the effectiveness of carbon dioxide as a greenhouse gas decreases
exponentially (Figure 1). This exponential
decline in the effectiveness of carbon dioxide as a
greenhouse gas is not contested by the believers in
global warming; they simply ignore it.
There is an experiment anyone can do to understand
this principle: Take a sheet of paper, and place it
over a window with sunlight coming through. You ...."
http://larouchepub.com/lar/Articles_2009/Articles_2009/Cap_and_Trade.pdf
http://larouchepub.com/lar/Articles_2009/Where_Punt_sp09.pdf

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